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This generalizes to the notion of abelian scheme ; a group scheme G over a base S is abelian if the structural morphism from G to S is proper and smooth with geometrically connected fibers They are automatically projective, and they have many applications, e. g., in geometric class field theory and throughout algebraic geometry.
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generalizes and notion
Model theory generalizes the notion of algebraic extension to arbitrary theories: an embedding of M into N is called an algebraic extension if for every x in N there is a formula p with parameters in M, such that p ( x ) is true and the set
The Hausdorff dimension generalizes the notion of the dimension of a real vector space.
The notion of irreducible fraction generalizes to the field of fractions of any unique factorization domain: any element of such a field can be written as a fraction in which denominator and numerator are coprime, by dividing both by their greatest common divisor.
However, the principle of special relativity generalizes the notion of inertial frame to include all physical laws, not simply Newton's first law.
This principle generalizes the notion of an inertial frame.
The dual notion of a colimit generalizes constructions such as disjoint unions, direct sums, coproducts, pushouts and direct limits.
The definition in the previous section generalizes the notion of inverse in group relative to the notion of identity.
The notion of paracompactness generalizes ordinary compactness ; a key motivation for the notion of paracompactness is that it is a sufficient condition for the existence of partitions of unity.
For this reason, the notion of a Noetherian ring generalizes such rings as.
This generalizes the notion of geodesic for Riemannian manifolds.
However, not every diagram commutes ( the notion of diagram strictly generalizes commutative diagram ): most simply, the diagram of a single object with an endomorphism (), or with two parallel arrows (, that is,, sometimes called the free quiver ), as used in the definition of equalizer need not commute.
He generalizes the notion in the following manner, arguing that " there are wider-scale institutional ' orders of interactionality ,' historically contingent yet structured.
In mathematics, a Cauchy net generalizes the notion of Cauchy sequence to nets defined on uniform spaces.
The notion of normal operators generalizes to an involutive algebra ; namely, an element x of an involutive algebra is said to be normal if.
* Two-point tensor, which generalizes this notion to tensors that have indices not just in the primal and dual space but in other vector spaces ( such as other tangent spaces on the same manifold ).
Note that x does not have to be an element of S. This concept profitably generalizes the notion of a limit and is the underpinning of concepts such as closed set and topological closure.
In combinatorics, a branch of mathematics, a matroid () or independence structure is a structure that captures and generalizes the notion of linear independence in vector spaces.
In the late 1960s matroid theorists asked for a more general notion that shares the different aspects of finite matroids and generalizes their duality.
* There is a general notion of monoid object in a monoidal category, which generalizes the ordinary notion of monoid.
The notion of " algebras for a monad " generalizes classical notions from universal algebra, and in this sense, monads can be thought of as " theories ".
In the latter case, the Gelfand-Naimark representation theorem is one avenue in the development of spectral theory for normal operators, and generalizes the notion of diagonalizing a normal matrix.
In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple ( more than 2 ) curves, and accounting properly for tangency.
generalizes and abelian
This further generalizes to the fact that in any finite abelian group, either the product of all elements is the identity, or there is precisely one element a of order 2 ( but not both ).
The solution for the hidden subgroup problem in the abelian case generalizes to finding the normal core in case of subgroups of arbitrary groups.
This definition generalizes to direct sums of finitely many abelian groups.
generalizes and scheme
Note, however, that while the notion of an association scheme generalizes the notion of a group, the notion of a commutative association scheme only generalizes the notion of a commutative group.
generalizes and ;
Compactness generalizes many important properties of closed and bounded intervals in the real line ; that is, intervals of the form for real numbers and.
* Formally, when working over the reals, as here, this is accomplished by considering the limit as ε → 0 ; but the " infinitesimal " language generalizes directly to Lie groups over general rings.
The multi-sentence version of the liar paradox generalizes to any circular sequence of such statements ( wherein the last statement asserts the truth / falsity of the first statement ), provided there are an odd number of statements asserting the falsity of their successor ; the following is a three-sentence version, with each statement asserting the falsity of its successor:
Thinking of matrices as tensors, the tensor rank generalizes to arbitrary tensors ; note that for tensors of order greater than 2 ( matrices are order 2 tensors ), rank is very hard to compute, unlike for matrices.
The ordinary generating function of a sequence can be expressed as a rational function ( the ratio of two polynomials ) if and only if the sequence is a linear recursive sequence ; this generalizes the examples above.
where ζ ( s, q ) is the Hurwitz zeta ; thus, in a certain sense, the Hurwitz zeta generalizes the Bernoulli polynomials to non-integer values of n.
If tighter bounds on the sectional curvature are known, then this property generalizes to give a comparison theorem between geodesic triangles in M and those in a suitable simply connected space form ; see Toponogov's theorem.
This theorem is a statement of the first isomorphism theorem of algebra to the case of vector spaces ; it generalizes to the splitting lemma.
The above definition is based in set theory ; the category-theoretic definition generalizes this into a functor from the free quiver to the category of sets.
; Metric multidimensional scaling: A superset of classical MDS that generalizes the optimization procedure to a variety of loss functions and input matrices of known distances with weights and so on.
This method obviously generalizes to any disjoint union of sets of intertermines ; the lexicographic order can be so obtained from the singleton sets
The cup product is a binary ( 2-ary ) operation ; one can define a ternary ( 3-ary ) and higher order operation called the Massey product, which generalizes the cup product.
The geometric median of a set of points is the point minimizing the sum of distances to the points in the set ; it generalizes the one-dimensional median and has been studied both from the point of view of facility location and robust statistics.
Note that there was nothing particular to two dimensions in the above presentation ; the above trivially generalizes to arbitrary dimensions.
generalizes and group
* Redundant array of independent disks ( RAID )-This method generalizes the device mirroring above by allowing one device in a group of N devices to fail and be replaced with content restored ( Device mirroring is RAID with N = 2 ).
* Redundant array of independent disks ( RAID )-This method generalizes the device mirroring above by allowing one device in a group of N devices to fail and be replaced with content restored ( Device mirroring is RAID with N = 2 ).
The above development generalizes in a more-or-less straightforward fashion to general principal G-bundles for some arbitrary Lie group G taking the place of U ( 1 ).
This generalizes the concept of centralizer in group theory.
In mathematics, the braid group on n strands, denoted by B < sub > n </ sub >, is a group which has an intuitive geometrical representation, and in a sense generalizes the symmetric group S < sub > n </ sub >.
Hyman Bass provided this definition, which generalizes the group of units of a ring: K < sub > 1 </ sub >( A ) is the abelianization of the infinite general linear group:
This construction generalizes the usual construction of the unitary group from the general linear group.
The direct sum of group representations generalizes the direct sum of the underlying modules, adding a group action to it.
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