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An order unit is an element of an ordered vector space which can be used to bound all elements from above. [1] In this way (as seen in the first example below) the order unit generalizes the unit element in the reals. According to H. H. Schaefer, "most of the ordered vector spaces occurring in analysis do not have order units." [2]
The mathematical derivation of an idealized Hubble's law for a uniformly expanding universe is a fairly elementary theorem of geometry in 3-dimensional Cartesian/Newtonian coordinate space, which, considered as a metric space, is entirely homogeneous and isotropic (properties do not vary with location or direction). Simply stated, the theorem ...
Tegmark's MUH is the hypothesis that our external physical reality is a mathematical structure. [3] That is, the physical universe is not merely described by mathematics, but is mathematics — specifically, a mathematical structure.
If U is Urysohn universal and X is any separable metric space, then there exists an isometric embedding f:X → U. (Other spaces share this property: for instance, the space l ∞ of all bounded real sequences with the supremum norm admits isometric embeddings of all separable metric spaces ("Fréchet embedding"), as does the space C[0,1] of all continuous functions [0,1]→R, again with the ...
Another important property of string theory is its supersymmetry, which together with extra dimensions are the two main proposals for resolving the hierarchy problem of the standard model, which is (roughly) the question of why gravity is so much weaker than any other force. The extra-dimensional solution involves allowing gravity to propagate ...
The last theorem was generalized by Lipscomb to the class of metric spaces of weight, >: There exist a one-dimensional metric space such that the subspace of + consisting of set of points, at most of whose coordinates are "rational" (suitably defined), is universal for the class of metric spaces whose Lebesgue covering dimension is less than ...
The unit of measurement used is the light-year (distance traveled by light in one Julian year; approximately 9.46 trillion kilometres). This list includes superclusters, galaxy filaments and large quasar groups (LQGs). The structures are listed based on their longest dimension.
Swedish astronomer Knut Lundmark was the first person to find observational evidence for expansion, in 1924. According to Ian Steer of the NASA/IPAC Extragalactic Database of Galaxy Distances, "Lundmark's extragalactic distance estimates were far more accurate than Hubble's, consistent with an expansion rate (Hubble constant) that was within 1% ...