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A hypersurface in a (Euclidean, affine, or projective) space of dimension two is a plane curve. In a space of dimension three, it is a surface. For example, the equation + + + = defines an algebraic hypersurface of dimension n − 1 in the Euclidean space of dimension n.
The exact solution of the minimum-diameter spanning tree problem, in the Euclidean plane, can be sped up from () to / + (), at the expense of using complicated range search data structures. The same method extends to higher dimensions, with smaller reductions in the exponent compared to the cubic algorithm.
For a graph with E edges and V vertices, Kruskal's algorithm can be shown to run in time O(E log E) time, with simple data structures. Here, O expresses the time in big O notation , and log is a logarithm to any base (since inside O -notation logarithms to all bases are equivalent, because they are the same up to a constant factor).
The technical statement appearing in Nash's original paper is as follows: if M is a given m-dimensional Riemannian manifold (analytic or of class C k, 3 ≤ k ≤ ∞), then there exists a number n (with n ≤ m(3m+11)/2 if M is a compact manifold, and with n ≤ m(m+1)(3m+11)/2 if M is a non-compact manifold) and an isometric embedding ƒ: M → R n (also analytic or of class C k). [15]
A kinetic Euclidean minimum spanning tree is a kinetic data structure that maintains the Euclidean minimum spanning tree (EMST) of a set P of n points that are moving continuously. For the set of points P in 2-dimensional space, there are two kinetic algorithms for maintenance of the EMST.
Note that if and : + is a smooth hypersurface immersion whose second fundamental form is positive, then the Gauss map: is a diffeomorphism, and so one knows from the start that is diffeomorphic to and, from elementary differential topology, that all immersions considered above are embeddings.
Tarski's axioms are an axiom system for Euclidean geometry, specifically for that portion of Euclidean geometry that is formulable in first-order logic with identity (i.e. is formulable as an elementary theory). As such, it does not require an underlying set theory. The only primitive objects of the system are "points" and the only primitive ...
In mathematics, a quadric or quadric hypersurface is the subspace of N-dimensional space defined by a polynomial equation of degree 2 over a field. Quadrics are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than affine space. An example is the quadric surface =