Definition & Betydelse | Engelska ordet HYPERSURFACE

Definition av HYPERSURFACE

1. (matematik) hyperyta

Exempel på hur man kan använda HYPERSURFACE i en mening

• Just as the perimeter of the square consists of four edges and the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eight cubical cells, meeting at right angles.
• Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is one less than that of the ambient space.
• In relativistic cosmology, Weyl's postulate stipulates that in the Friedmann model of the universe (a fluid cosmological model), the wordlines of fluid particles (modeling galaxies) should be hypersurface orthogonal.
• In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas).
• More generally, a smooth quadric (degree 2) hypersurface X of any dimension n is rational, by stereographic projection.
• The special case for a hypersurface (an (n-1)-dimensional submanifolds in an n-dimensional Euclidean space) was proved by H.
• Both contact and symplectic geometry are motivated by the mathematical formalism of classical mechanics, where one can consider either the even-dimensional phase space of a mechanical system or constant-energy hypersurface, which, being codimension one, has odd dimension.
• A single equation defines a hypersurface, and simultaneous Diophantine equations give rise to a general algebraic variety V over K; the typical question is about the nature of the set V(K) of points on V with co-ordinates in K, and by means of height functions, quantitative questions about the "size" of these solutions may be posed, as well as the qualitative issues of whether any points exist, and if so whether there are an infinite number.
• A corollary of this theorem is that, if two irreducible polynomials (or more generally two square-free polynomials) define the same hypersurface, then one is the product of the other by a nonzero constant.
• Invariantly, a characteristic hypersurface is a hypersurface whose conormal bundle is in the characteristic set of P.
• Geometrically, an isotropic line of the quadratic form corresponds to a point of the associated quadric hypersurface in projective space.
• The microcanonical ensemble is very natural from the naïve physical point of view: this is just a natural equidistribution on the isoenergetic hypersurface.
• In a four-dimensional spacetime manifold, a hypersurface is a three-dimensional submanifold that can be either timelike, spacelike, or null.
• Two rings of ten pentagonal antiprisms each bound the hypersurface of the four-dimensional grand antiprism.
• The notion of a CR structure attempts to describe intrinsically the property of being a hypersurface (or certain real submanifolds of higher codimension) in complex space by studying the properties of holomorphic vector fields which are tangent to the hypersurface.
• A particular application of their results is a Bernstein theorem for closed spacelike hypersurfaces of Minkowski space whose mean curvature is zero; any such hypersurface must be a plane.
• Notable contributions include the theory of matrix factorizations for maximal Cohen–Macaulay modules over hypersurface rings, the Eisenbud–Goto conjecture on degrees of generators of syzygy modules, and the Buchsbaum–Eisenbud criterion for exactness of a complex.
• In particular it allows the total of matter plus the gravitating energy–momentum to form a conserved current within the framework of general relativity, so that the total energy–momentum crossing the hypersurface (3-dimensional boundary) of any compact space–time hypervolume (4-dimensional submanifold) vanishes.
• In hyperbolic geometry, a horosphere (or parasphere) is a specific hypersurface in hyperbolic n-space.
• In 2008, Walter Craig and Steven Weinstein proved that under a nonlocal constraint, the initial value problem is well-posed for initial data given on a codimension-one hypersurface.

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