![]() Optical simulations of gravitational effects in the Newton–Schrödinger system. Control of light by curved space in nanophotonic structures. Transformation optics that mimics the system outside a Schwarzschild black hole. Mimicking celestial mechanics in metamaterials. Metric signature transitions in optical metamaterials. Linear and nonlinear optics in curved space. General relativity in electrical engineering. This paper represents a step towards on-chip quantum simulation of materials science and interacting particles in curved space.Ĭannon, J. We present a proof-of-principle experimental realization of one such lattice. We present numerical simulations of hyperbolic analogues of the kagome lattice that show unusual densities of states in which a macroscopic number of degenerate eigenstates comprise a spectrally isolated flat band. We show that networks of coplanar waveguide resonators can create a class of materials that constitute lattices in an effective hyperbolic space with constant negative curvature. Here we make use of the previously overlooked property that these lattice sites are deformable and permit tight-binding lattices that are unattainable even in solid-state systems. Lattices of coplanar waveguide resonators constitute artificial materials for microwave photons, in which interactions between photons can be incorporateded either through the use of nonlinear resonator materials or through coupling between qubits and resonators. The abbreviations sh, ch, th, cth are also currently used, depending on personal preference.After two decades of development, cavity quantum electrodynamics with superconducting circuits has emerged as a rich platform for quantum computation and simulation. Lambert adopted the names, but altered the abbreviations to those used today. ( sinus/cosinus hyperbolico) to refer to hyperbolic functions. ( sinus/cosinus circulare) to refer to circular functions and Sh. Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and Johann Heinrich Lambert. As a result, the other hyperbolic functions are meromorphic in the whole complex plane.īy Lindemann–Weierstrass theorem, the hyperbolic functions have a transcendental value for every non-zero algebraic value of the argument. The hyperbolic sine and the hyperbolic cosine are entire functions. In complex analysis, the hyperbolic functions arise when applying the ordinary sine and cosine functions to an imaginary angle. The hyperbolic functions may be defined in terms of the legs of a right triangle covering this sector. The size of a hyperbolic angle is twice the area of its hyperbolic sector. The hyperbolic functions take a real argument called a hyperbolic angle. For points on the hyperbola below the x-axis, the area is considered negative (see animated version with comparison with the trigonometric (circular) functions). area hyperbolic cosine " arcosh" (also denoted " cosh −1", " acosh" or sometimes " arccosh")Ī ray through the unit hyperbola x 2 − y 2 = 1 at the point (cosh a, sinh a), where a is twice the area between the ray, the hyperbola, and the x-axis.area hyperbolic sine " arsinh" (also denoted " sinh −1", " asinh" or sometimes " arcsinh").hyperbolic cotangent " coth" ( / ˈ k ɒ θ, ˈ k oʊ θ/), Ĭorresponding to the derived trigonometric functions.hyperbolic secant " sech" ( / ˈ s ɛ tʃ, ˈ ʃ ɛ k/),.hyperbolic cosecant " csch" or " cosech" ( / ˈ k oʊ s ɛ tʃ, ˈ k oʊ ʃ ɛ k/ ).hyperbolic tangent " tanh" ( / ˈ t æ ŋ, ˈ t æ n tʃ, ˈ θ æ n/),.hyperbolic cosine " cosh" ( / ˈ k ɒ ʃ, ˈ k oʊ ʃ/),.hyperbolic sine " sinh" ( / ˈ s ɪ ŋ, ˈ s ɪ n tʃ, ˈ ʃ aɪ n/),.Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity. They also occur in the solutions of many linear differential equations (such as the equation defining a catenary), cubic equations, and Laplace's equation in Cartesian coordinates. Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry. Also, similarly to how the derivatives of sin( t) and cos( t) are cos( t) and –sin( t) respectively, the derivatives of sinh( t) and cosh( t) are cosh( t) and +sinh( t) respectively. Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola. In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle.
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