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Round-trip translation (RTT), also known as back-and-forth translation, recursive translation and bi-directional translation, is the process of translating a word, phrase or text into another language (forward translation), then translating the result back into the original language (back translation), using machine translation (MT) software.
2-sphere Commonly simply called a sphere. For its complex structure, see Riemann sphere. Homeomorphic to the complex projective line 3-sphere Parallelizable, principal U(1)-bundle over the -sphere, Lie group structure Sp(1) = SU(2). 4-sphere Homeomorphic to the quaternionic projective line, .
DeepL Translator is a neural machine translation service that was launched in August 2017 and is owned by Cologne-based DeepL SE. The translating system was first developed within Linguee and launched as entity DeepL .
The four Euclidean coordinates for S 3 are redundant since they are subject to the condition that x 0 2 + x 1 2 + x 2 2 + x 3 2 = 1. As a 3-dimensional manifold one should be able to parameterize S 3 by three coordinates, just as one can parameterize the 2-sphere using two coordinates (such as latitude and longitude).
Consider the linear subspace of the n-dimensional Euclidean space R n that is spanned by a collection of linearly independent vectors , …,. To find the volume element of the subspace, it is useful to know the fact from linear algebra that the volume of the parallelepiped spanned by the is the square root of the determinant of the Gramian matrix of the : (), = ….
The n-sphere of unit radius centered at the origin is denoted S n and is often referred to as "the" n-sphere. The ordinary sphere is a 2-sphere, because it is a 2-dimensional surface which is embedded in 3-dimensional space. In topology, the n-sphere is an example of a compact topological manifold without boundary.
A Morin surface seen from "above" Sphere eversion process as described in [1] Paper sphere eversion and Morin surface Paper Morin surface (sphere eversion halfway) with hexagonal symmetry In differential topology , sphere eversion is a theoretical process of turning a sphere inside out in a three-dimensional space (the word eversion means ...
where S n − 1 (r) is an (n − 1)-sphere of radius r (being the surface of an n-ball of radius r) and dA is the area element (equivalently, the (n − 1)-dimensional volume element). The surface area of the sphere satisfies a proportionality equation similar to the one for the volume of a ball: If A n − 1 ( r ) is the surface area of an ( n ...