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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.
In the hypersurface case where =, singularities occur only for . An example of such singular solution of the Plateau problem is the Simons cone , a cone over S 3 × S 3 {\displaystyle S^{3}\times S^{3}} in R 8 {\displaystyle \mathbb {R} ^{8}} that was first described by Jim Simons and was shown to be an area minimizer by Bombieri , De Giorgi ...
The Euclidean algorithm was probably invented before Euclid, depicted here holding a compass in a painting of about 1474. The Euclidean algorithm is one of the oldest algorithms in common use. [27] It appears in Euclid's Elements (c. 300 BC), specifically in Book 7 (Propositions 1–2) and Book 10 (Propositions 2–3). In Book 7, the algorithm ...
The Gauss map can be defined for hypersurfaces in R n as a map from a hypersurface to the unit sphere S n − 1 ⊆ R n. For a general oriented k - submanifold of R n the Gauss map can also be defined, and its target space is the oriented Grassmannian G ~ k , n {\displaystyle {\tilde {G}}_{k,n}} , i.e. the set of all oriented k -planes in R n .
In geometry, a hyperplane of an n-dimensional space V is a subspace of dimension n − 1, or equivalently, of codimension 1 in V.The space V may be a Euclidean space or more generally an affine space, or a vector space or a projective space, and the notion of hyperplane varies correspondingly since the definition of subspace differs in these settings; in all cases however, any hyperplane can ...
In a statistical-classification problem with two classes, a decision boundary or decision surface is a hypersurface that partitions the underlying vector space into two sets, one for each class. The classifier will classify all the points on one side of the decision boundary as belonging to one class and all those on the other side as belonging ...
Lamé's Theorem is the result of Gabriel Lamé's analysis of the complexity of the Euclidean algorithm.Using Fibonacci numbers, he proved in 1844 [1] [2] that when looking for the greatest common divisor (GCD) of two integers a and b, the algorithm finishes in at most 5k steps, where k is the number of digits (decimal) of b.
A simple example of mean curvature flow is given by a family of concentric round hyperspheres in +. The mean curvature of an m {\displaystyle m} -dimensional sphere of radius R {\displaystyle R} is H = m / R {\displaystyle H=m/R} .