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In uniform scaling with a non-zero scale factor, all non-zero vectors retain their direction (as seen from the origin), or all have the direction reversed, depending on the sign of the scaling factor. In non-uniform scaling only the vectors that belong to an eigenspace will retain their direction. A vector that is the sum of two or more non ...
The simplest of these is called elliptic geometry and it is considered a non-Euclidean geometry due to its lack of parallel lines. [12] By formulating the geometry in terms of a curvature tensor, Riemann allowed non-Euclidean geometry to apply to higher dimensions. Beltrami (1868) was the first to apply Riemann's geometry to spaces of negative ...
Scaling (geometry) a similar notion in vector spaces Homothetic center , the center of a homothetic transformation taking one of a pair of shapes into the other The Hadwiger conjecture on the number of strictly smaller homothetic copies of a convex body that may be needed to cover it
In contrast, angles and ratios are not invariant under non-uniform scaling (such as stretching). The sum of a triangle's interior angles (180°) is invariant under all the above operations. As another example, all circles are similar: they can be transformed into each other and the ratio of the circumference to the diameter is invariant ...
A quasi-isometry is a map that preserves the "large-scale structure" of a metric space. Quasi-isometries need not be continuous. Quasi-isometries need not be continuous. For example, R 2 {\displaystyle \mathbb {R} ^{2}} and its subspace Z 2 {\displaystyle \mathbb {Z} ^{2}} are quasi-isometric, even though one is connected and the other is discrete.
More complicated non-algebraic structures combine an algebraic component and a non-algebraic component. For example, the structure of a topological group consists of a topology and the structure of a group. Thus it belongs to the product of P(P(X)) and another ("algebraic") set in the scale; this product is again a set in the scale.
An indecomposable module is a non-zero module that cannot be written as a direct sum of two non-zero submodules. Every simple module is indecomposable, but there are indecomposable modules that are not simple (e.g. uniform modules). Faithful A faithful module M is one where the action of each r ≠ 0 in R on M is nontrivial (i.e. r ⋅ x ≠ 0 ...
Large-scale properties of a space—such as boundedness, or the degrees of freedom of the space—do not depend on such features. Coarse geometry and coarse topology provide tools for measuring the large-scale properties of a space, and just as a metric or a topology contains information on the small-scale structure of a space, a coarse ...