from Wiktionary, Creative Commons Attribution/Share-Alike License
- n. A manifold which is a subset of another, so that the inclusion function is an embedding or, sometimes, an immersion
from The Century Dictionary and Cyclopedia
- n. The selected elements of a partitioned manifold.
Sorry, no etymologies found.
This would include: the formation and decay of (nonsupersymmetric) black holes; graviton scattering at hyper-Planckian energies; physically continuous transitions from one topology, through a quantum state with no geometric interpretation, to a different topology; states where a submanifold of spacetime has a noncommutative geometry; states with a variety of apparent geometric singularities, where the physics is nonsingular.
The general hypothesis is that the universe as a whole is 3+1+d dimensional (so that there are d extra, spatial dimensions), with gravity propagating in all dimensions, but the standard model fields confined to a 3+1 dimensional submanifold that comprises our observable universe.
In this model, matter fields are confined to a four-dimensional submanifold and gravity is described by two different terms; one five-dimensional gravitational action and another four-dimensional action.
This submanifold is called the brane (as in membrane).
The general hypothesis is that the universe as a whole is 3+1+d dimensional (so that there are d extra spatial dimensions), with gravity propagating in all dimensions, but the standard model fields confined to a 3+1 dimensional submanifold that comprises our observable universe.
There are also other, field-theoretic, ways of localizing matter on a submanifold, by using a topological defect.
Start with a three-manifold Î£ equipped with quantities which, when Î£ is embedded as a spacelike submanifold of spacetime, become initial data for the vacuum field equations.
In their version of manifold learning, they assume that the space of target functions (e.g. all possible classifiers), consists of functions supported on a submanifold † † of the original high-dimensional euclidean/feature space.
It’s really just a topological surface, and its action functional is just its area as a submanifold of spacetime.
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