Search results
Results from the WOW.Com Content Network
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.
Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency.The frequency representation is found by using the Fourier transform for functions on unbounded domains such as the full real line or by Fourier series for functions on bounded domains, especially periodic functions on finite intervals.
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 ...
On a Riemann surface the Poincaré lemma states that every closed 1-form or 2-form is locally exact. [2] Thus if ω is a smooth 1-form with dω = 0 then in some open neighbourhood of a given point there is a smooth function f such that ω = df in that neighbourhood; and for any smooth 2-form Ω there is a smooth 1-form ω defined in some open neighbourhood of a given point such that Ω = dω ...
The search engine that helps you find exactly what you're looking for. Find the most relevant information, video, images, and answers from all across the Web.
If harmonic Maass forms are interpreted as harmonic sections of the line bundle of modular forms of weight equipped with the Petersson metric over the modular curve, then this differential operator can be viewed as a composition of the Hodge star operator and the antiholomorphic differential. The notion of harmonic Maass forms naturally ...
AOL Mail welcomes Verizon customers to our safe and delightful email experience!
In mathematics, a harmonic morphism is a (smooth) map : (,) (,) between Riemannian manifolds that pulls back real-valued harmonic functions on the codomain to harmonic functions on the domain. Harmonic morphisms form a special class of harmonic maps , namely those that are horizontally (weakly) conformal.