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In particular, a C k-atlas that is C 0-compatible with a C 0-atlas that defines a topological manifold is said to determine a C k differential structure on the topological manifold. The C k equivalence classes of such atlases are the distinct C k differential structures of the manifold. Each distinct differential structure is determined by a ...
The study of dynamic equations on time scales reveals such discrepancies, and helps avoid proving results twice—once for differential equations and once again for difference equations. The general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so-called time scale (also known as a time-set ...
A topological manifold that is in the image of is said to "admit a differentiable structure", and the fiber over a given topological manifold is "the different differentiable structures on the given topological manifold". Thus given two categories, the two natural questions are:
Most differentiable programming frameworks work by constructing a graph containing the control flow and data structures in the program. [7] Attempts generally fall into two groups: Static, compiled graph-based approaches such as TensorFlow, [note 1] Theano, and MXNet.
As another example, there can be no local diffeomorphism from the 2-sphere to Euclidean 2-space, although they do indeed have the same local differentiable structure. This is because all local diffeomorphisms are continuous , the continuous image of a compact space is compact, and the 2-sphere is compact whereas Euclidean 2-space is not.
By Leah Douglas and Julie Steenhuysen (Reuters) -California's public health department reported a possible case of bird flu in a child with mild respiratory symptoms on Tuesday, but said there was ...
The San Francisco 49ers suspended linebacker De’Vondre Campbell for the rest of the regular season after he refused to enter a game after losing his starting job. General manager John Lynch ...
Bill Murray appeared on Travis and Jason Kelce’s “New Heights” podcast (via The Daily Beast) and defended the current cast of “Saturday Night Live” from critics who say the show has ...