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In an area of mathematics called differential topology, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere. That is, M is a sphere from the point of view of all its topological properties, but carrying a smooth structure that is not the familiar one (hence the name ...
Diffeomorphometry is the metric study of imagery, shape and form in the discipline of computational anatomy (CA) in medical imaging.The study of images in computational anatomy rely on high-dimensional diffeomorphism groups which generate orbits of the form {}, in which images can be dense scalar magnetic resonance or computed axial tomography images.
A diffeomorphic mapping system is a system designed to map, manipulate, and transfer information which is stored in many types of spatially distributed medical imagery. Diffeomorphic mapping is the underlying technology for mapping and analyzing information measured in human anatomical coordinate systems which have been measured via Medical ...
Blender supports Python scripting for the creation of custom tools, prototyping, importing/exporting from other formats, and task automation. This allows for integration with several external render engines through plugins/addons. Blender itself can also be compiled & imported as a python library for further automation and development.
Testing whether a differentiable map is a diffeomorphism can be made locally under some mild restrictions. This is the Hadamard-Caccioppoli theorem: [1] If , are connected open subsets of such that is simply connected, a differentiable map : is a diffeomorphism if it is proper and if the differential: is bijective (and hence a linear isomorphism) at each point in .
If the Gaussian curvature of a surface M is everywhere positive, then the Euler characteristic is positive so M is homeomorphic (and therefore diffeomorphic) to S 2. If in addition the surface is isometrically embedded in E 3 , the Gauss map provides an explicit diffeomorphism.
Diffeomorphic Dimensionality Reduction or Diffeomap [31] learns a smooth diffeomorphic mapping which transports the data onto a lower-dimensional linear subspace. The methods solves for a smooth time indexed vector field such that flows along the field which start at the data points will end at a lower-dimensional linear subspace, thereby ...
Furthermore, this compact manifold with boundary, which is known as the Milnor fiber (of the isolated singular point of at the origin), is diffeomorphic to the intersection of the closed (+)-ball (bounded by the small (+)-sphere) with the (non-singular) hypersurface where = and is any sufficiently small non-zero complex number.
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