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In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold M using partial differential equations.The key observation is that, given a Riemannian metric on M, every cohomology class has a canonical representative, a differential form that vanishes under the Laplacian operator of the metric.
A harmonic differential (one-form) is precisely the real part of an (analytic) complex differential. [1]: 172 To prove this one shows that u + iv satisfies the Cauchy–Riemann equations exactly when u + iv is locally an analytic function of x + iy. Of course an analytic function w(z) = u + iv is the local derivative of something (namely ∫w(z ...
The codifferential is the adjoint of the exterior derivative with respect to the square-integrable scalar product: , = , , where is a -form and a ()-form. This property is useful as it can be used to define the codifferential even when the manifold is non-orientable (and the Hodge star operator not defined).
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Four ordered points on a projective range are called harmonic points when there is a tetrastigm in the plane such that the first and third are codots and the other two points are on the connectors of the third codot. [6] If p is a point not on a straight with harmonic points, the joins of p with the points are harmonic straights.
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The first nonlinear optical effect to be predicted was two-photon absorption, by Maria Goeppert Mayer for her PhD in 1931, but it remained an unexplored theoretical curiosity until 1961 and the almost simultaneous observation of two-photon absorption at Bell Labs [4] and the discovery of second-harmonic generation by Peter Franken et al. at University of Michigan, both shortly after the ...
The Euler-Lagrange equation of a Dirichlet form is a non-local analogue of an elliptic equations in divergence form. Equations of this type are studied using variational methods and they are expected to satisfy similar properties. [2] [3] [4]