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- There are several different systems of axioms for affine space.
- There are two ways to formally define affine planes, which are equivalent for affine planes over a field.
- Here are all of the Dynkin graphs for affine groups up to 10 nodes.
- All results stated for a regular polygon also hold for affine-regular polygons; in particular, results concerning the unit square also apply to other parallelograms, including rectangles and rhombuses.
- For affine and projective varieties, the singularities are the points where the Jacobian matrix has a rank which is lower than at other points of the variety.
- The second motivation for affine connections comes from the notion of a covariant derivative of vector fields.
- For example, for affine transformation the registration from one brain scan to another may be found with the alignlinear program and written to the special air-file that stores the transformation matrix.
- Affine shape adaptation can also be used for affine invariant texture recognition and affine invariant texture segmentation.
- For affine A n groups, the double affine braid group is the fundamental group of the space of n distinct points on a 2-dimensional torus.
- The SIFT-Rank descriptor was shown to improve the performance of the standard SIFT descriptor for affine feature matching.
- There is also a heap-like structure based on the dynamic convex hull data structure which achieves better performance for affine motion of the priorities, but doesn't support curved trajectories.
- However, these facts are counterintuitive: we do not normally expect open sets, other than connected components, to be compact, and for affine varieties (for example, Euclidean space) we do not even expect the space itself to be compact.
- The projective Zariski topology is defined for projective algebraic sets just as the affine one is defined for affine algebraic sets, by taking the subspace topology.
- All of these concepts are usually defined as subsets of an ambient vector space (except for affine spaces, which are also considered as "vector spaces forgetting the origin"), rather than being axiomatized independently.
- Analogously to the definitions for affine and projective varieties, a k-variety is a variety defined over k if the stalk of the structure sheaf at the generic point is a regular extension of k; furthermore, every variety has a minimal field of definition.
- Ordered geometry is a fundamental geometry forming a common framework for affine, Euclidean, absolute, and hyperbolic geometry (but not for projective geometry).
- Young's lattice prominently figures in algebraic combinatorics, forming the simplest example of a differential poset in the sense of, It is also closely connected with the crystal bases for affine Lie algebras.
- Based on the work of Lawrence Edwards In a foundational sense, projective geometry and ordered geometry are elementary since they involve a minimum of axioms and either can be used as the foundation for affine and Euclidean geometry.
- In classical algebraic geometry (that is, the subject prior to the Grothendieck revolution of the late 1950s and 1960s) the Zariski topology was defined for affine and projective varieties.
- We identify as affine theorems any geometric result that is invariant under the affine group (in Felix Klein's Erlangen programme this is its underlying group of symmetry transformations for affine geometry).