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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.
In harmonic analysis, a field within mathematics, Littlewood–Paley theory is a theoretical framework used to extend certain results about L 2 functions to L p functions for 1 < p < ∞. It is typically used as a substitute for orthogonality arguments which only apply to L p functions when p = 2.
Hybrid homeschooling or flex-school [27] is a form of homeschooling in which children split their time between homeschool and a more traditional schooling environment like a school. [61] The number of students who participated in hybrid homeschooling increased during the COVID-19 pandemic .
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 ...
at . Ate Food Journal “If you prefer keeping a visual food diary, Ate is the best option,” White says. “You can snap a picture of your food and the app will help you to look back on the food ...
Santiago Felipe/Getty, Giles Keyte/Universal Pictures 'Wicked' director Jon M. Chu; 'Wicked' stars Cynthia Erivo and Ariana Grande
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 ...
Accused murderer Luigi Mangione’s grandmother reportedly left tens of millions of dollars to her children and grandchildren after she died, but did so on one condition: that any grandchild ...
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