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Affine Motion Estimation
Affine may describe any of various topics concerned with connections or affinities. It may refer to: * Affine, a relative by marriage in law and anthropology * Affine cipher, a special case of the more general substitution cipher * Affine combination, a certain kind of constrained linear combination * Affine connection, a connection on the tangent bundle of a differentiable manifold * Affine Coordinate System, a coordinate system that can be viewed as a Cartesian coordinate system where the axes have been placed so that they are not necessarily orthogonal to each other. See tensor. * Affine differential geometry, a geometry that studies differential invariants under the action of the special affine group * Affine gap penalty, the most widely used scoring function used for sequence alignment, especially in bioinformatics * Affine geometry, a geometry characterized by parallel lines * Affine group, the group of all invertible affine transformations from any affine space over ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine
Affine may describe any of various topics concerned with connections or affinities. It may refer to: * Affine, a Affinity_(law)#Terminology, relative by marriage in law and anthropology * Affine cipher, a special case of the more general substitution cipher * Affine combination, a certain kind of constrained linear combination * Affine connection, a connection on the tangent bundle of a differentiable manifold * Affine Coordinate System, a coordinate system that can be viewed as a Cartesian coordinate system where the axes have been placed so that they are not necessarily orthogonal to each other. See tensor. * Affine differential geometry, a geometry that studies differential invariants under the action of the special affine group * Gap penalty#Affine, Affine gap penalty, the most widely used scoring function used for sequence alignment, especially in bioinformatics * Affine geometry, a geometry characterized by parallel lines * Affine group, the group of all invertible affine ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Logic
Affine logic is a substructural logic whose proof theory rejects the structural rule of contraction. It can also be characterized as linear logic with weakening. The name "affine logic" is associated with linear logic, to which it differs by allowing the weakening rule. Jean-Yves Girard introduced the name as part of the geometry of interaction semantics of linear logic, which characterizes linear logic in terms of linear algebra; here he alludes to affine transformations on vector spaces. Affine logic predated linear logic. V. N. Grishin used this logic in 1974, after observing that Russell's paradox cannot be derived in a set theory without contraction, even with an unbounded comprehension axiom.Cf. Frederic Fitch's demonstrably consistent set theory Likewise, the logic formed the basis of a decidable sub-theory of predicate logic, called 'Direct logic' (Ketonen & Wehrauch, 1984; Ketonen & Bellin, 1989). Affine logic can be embedded into linear logic by rewri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, '' affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generally, an affine transformation is an automorphism of an affine space (Euclidean spaces are specific affine spaces), that is, a function which maps an affine space onto itself while preserving both the dimension of any affine subspaces (meaning that it sends points to points, lines to lines, planes to planes, and so on) and the ratios of the lengths of parallel line segments. Consequently, sets of parallel affine subspaces remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. If is the point set of an affine space, then every affine transformation on can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Space
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments. Affine space is the setting for affine geometry. As in Euclidean space, the fundamental objects in an affine space are called '' points'', which can be thought of as locations in the space without any size or shape: zero-dimensional. Through any pair of points an infinite straight line can be drawn, a one-dimensional set of points; through any three points that are not collinear, a two-dimensional plane can be drawn; and, in general, through points in general position, a -dimensional flat or affine subspace can be drawn. Affine space is characterized by a notion of pairs of parallel lines that lie within the same plane but never meet each-other (non-parallel lines wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morphism Of Schemes
In algebraic geometry, a morphism of schemes generalizes a morphism of algebraic varieties just as a scheme generalizes an algebraic variety. It is, by definition, a morphism in the category of schemes. A morphism of algebraic stacks generalizes a morphism of schemes. Definition By definition, a morphism of schemes is just a morphism of locally ringed spaces. Isomorphisms are defined accordingly. A scheme, by definition, has open affine charts and thus a morphism of schemes can also be described in terms of such charts (compare the definition of morphism of varieties). Let ƒ:''X''→''Y'' be a morphism of schemes. If ''x'' is a point of ''X'', since ƒ is continuous, there are open affine subsets ''U'' = Spec ''A'' of ''X'' containing ''x'' and ''V'' = Spec ''B'' of ''Y'' such that ƒ(''U'') ⊆ ''V''. Then ƒ: ''U'' → ''V'' is a morphism of affine schemes and thus is induced by some ring homomorphism ''B'' → ''A'' (cf. #Affine case.) In fact, one can use this des ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Morphism
In algebraic geometry, a sheaf of algebras on a ringed space ''X'' is a sheaf of commutative rings on ''X'' that is also a sheaf of \mathcal_X-modules. It is quasi-coherent if it is so as a module. When ''X'' is a scheme, just like a ring, one can take the global Spec of a quasi-coherent sheaf of algebras: this results in the contravariant functor \operatorname_X from the category of quasi-coherent (sheaves of) \mathcal_X-algebras on ''X'' to the category of schemes that are affine over ''X'' (defined below). Moreover, it is an equivalence: the quasi-inverse is given by sending an affine morphism f: Y \to X to f_* \mathcal_Y. Affine morphism A morphism of schemes In algebraic geometry, a morphism of schemes generalizes a morphism of algebraic varieties just as a scheme generalizes an algebraic variety. It is, by definition, a morphism in the category of schemes. A morphism of algebraic stacks generali ... f: X \to Y is called affine if Y has an open affine cover U ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Scheme
In commutative algebra, the prime spectrum (or simply the spectrum) of a commutative ring R is the set of all prime ideals of R, and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with a sheaf of rings. Zariski topology For any ideal I of R, define V_I to be the set of prime ideals containing I. We can put a topology on \operatorname(R) by defining the collection of closed sets to be :\big\. This topology is called the Zariski topology. A basis for the Zariski topology can be constructed as follows: For f\in R, define D_f to be the set of prime ideals of R not containing f. Then each D_f is an open subset of \operatorname(R), and \big\ is a basis for the Zariski topology. \operatorname(R) is a compact space, but almost never Hausdorff: In fact, the maximal ideals in R are precisely the closed points in this topology. By the same reasoning, \operatorname(R) is not, in general, a T1 space. However, \operat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Representation
In mathematics, an affine representation of a topological Lie group ''G'' on an affine space ''A'' is a continuous ( smooth) group homomorphism from ''G'' to the automorphism group of ''A'', the affine group Aff(''A''). Similarly, an affine representation of a Lie algebra g on ''A'' is a Lie algebra homomorphism from g to the Lie algebra aff(''A'') of the affine group of ''A''. An example is the action of the Euclidean group E(''n'') on the Euclidean space E''n''. Since the affine group in dimension ''n'' is a matrix group in dimension ''n'' + 1, an affine representation may be thought of as a particular kind of linear representation. We may ask whether a given affine representation has a fixed point in the given affine space ''A''. If it does, we may take that as origin and regard ''A'' as a vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Group
In mathematics, the affine group or general affine group of any affine space is the group of all invertible affine transformations from the space into itself. In the case of a Euclidean space (where the associated field of scalars is the real numbers), the affine group consists of those functions from the space to itself such that the image of every line is a line. Over any field, the affine group may be viewed as a matrix group in a natural way. If the associated field of scalars is the real or complex field, then the affine group is a Lie group. Relation to general linear group Construction from general linear group Concretely, given a vector space , it has an underlying affine space obtained by "forgetting" the origin, with acting by translations, and the affine group of can be described concretely as the semidirect product of by , the general linear group of : :\operatorname(V) = V \rtimes \operatorname(V) The action of on is the natural one (linear transformations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affinity (law)
In law and in cultural anthropology, affinity is the kinship relationship created or that exists between two people as a result of someone's marriage. It is the relationship each party in the marriage has to the family of the other party in the marriage. It does not cover the marital relationship itself. Laws, traditions and customs relating to affinity vary considerably, sometimes ceasing with the death of one of the marriage partners through whom affinity is traced, and sometimes with the divorce of the marriage partners. In addition to kinship by marriage, "affinity" can sometimes also include kinship by adoption or a step relationship. Unlike blood relationships ( consanguinity), which may have genetic consequences, affinity is essentially a social or moral construct, at times backed by legal consequences. In law, affinity may be relevant in relation to prohibitions on incestuous sexual relations and in relation to whether particular couples are prohibited from marryin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Geometry
In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric notions of distance and angle. As the notion of '' parallel lines'' is one of the main properties that is independent of any metric, affine geometry is often considered as the study of parallel lines. Therefore, Playfair's axiom (Given a line and a point not on , there is exactly one line parallel to that passes through .) is fundamental in affine geometry. Comparisons of figures in affine geometry are made with affine transformations, which are mappings that preserve alignment of points and parallelism of lines. Affine geometry can be developed in two ways that are essentially equivalent. In synthetic geometry, an affine space is a set of ''points'' to which is associated a set of lines, which satisfy some axioms (such as Playfair's axiom). Affine geometry can also be developed on the basis of linear algebra. In this context an affine ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gap Penalty
A Gap penalty is a method of scoring alignments of two or more sequences. When aligning sequences, introducing gaps in the sequences can allow an alignment algorithm to match more terms than a gap-less alignment can. However, minimizing gaps in an alignment is important to create a useful alignment. Too many gaps can cause an alignment to become meaningless. Gap penalties are used to adjust alignment scores based on the number and length of gaps. The five main types of gap penalties are constant, linear, affine, convex, and profile-based. Applications * Genetic sequence alignment - In bioinformatics, gaps are used to account for genetic mutations occurring from Insertion (genetics), insertions or Deletion (genetics), deletions in the sequence, sometimes referred to as ''indels''. Insertions or deletions can occur due to single mutations, unbalanced crossover in meiosis, slipped strand mispairing, and chromosomal translocation. The notion of a gap in an alignment is important in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
