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	~~Gabrielle~~ is ~~what her life partner loves to call your sweetheart though she doesn~~'~~t very like being called like that~~. ~~Fish dealing with acne~~ is ~~something her partner doesn~~'~~t really like nevertheless~~ , ~~she does~~. [~~http~~://~~managing~~.~~org~~/ ~~Managing~~] ~~people happens~~ to ~~be what she does although she plans~~ on ~~diaper changing it~~. For ~~years she~~'~~s been that reside~~ in ~~Massachusetts~~. ~~Go to her own website to find out more~~: http://~~circuspartypanama~~.~~com~~<br><br>~~Here~~ is ~~my blog post hack clash~~ of ~~clans (~~[~~http~~://~~circuspartypanama~~.~~com linked internet site~~])		{{one source\|date=December 2012}}
			In [[mathematics]], the '''smash product''' of two [[pointed space]]s (i.e. [[topological space]]s with distinguished basepoints) ''X'' and ''Y'' is the [[quotient space\|quotient]] of the [[product space]] ''X'' × ''Y'' under the identifications (''x'', ''y''<sub>0</sub>) ∼ (''x''<sub>0</sub>, ''y'') for all ''x'' ∈ ''X'' and ''y'' ∈ ''Y''. The smash product is usually denoted ''X'' ∧ ''Y''. The smash product depends on the choice of basepoints (unless both ''X'' and ''Y'' are [[homogeneous space\|homogeneous]]).

			One can think of ''X'' and ''Y'' as sitting inside ''X'' × ''Y'' as the [[subspace (topology)\|subspaces]] ''X'' × {''y''<sub>0</sub>} and {''x''<sub>0</sub>} × ''Y''. These subspaces intersect at a single point: (''x''<sub>0</sub>, ''y''<sub>0</sub>), the basepoint of ''X'' × ''Y''. So the union of these subspaces can be identified with the [[wedge sum]] ''X'' ∨ ''Y''. The smash product is then the quotient
			:<math>X \wedge Y = (X \times Y) / (X \vee Y). \, </math>

			The smash product has important applications in [[homotopy theory]], a branch of [[algebraic topology]]. In homotopy theory, one often works with a different [[category (mathematics)\|category]] of spaces than the category of all topological spaces. In some of these categories the definition of the smash product must be modified slightly. For example, the smash product of two [[CW complex]]es is a CW complex if one uses the product of CW complexes in the definition rather than the product topology. Similar modifications are necessary in other categories.

			==Examples==
			*The smash product of any pointed space ''X'' with a [[0-sphere]] is homeomorphic to ''X''.
			*The smash product of two circles is a quotient of the [[torus]] homeomorphic to the 2-sphere.
			*More generally, the smash product of two spheres ''S''<sup>''m''</sup> and ''S''<sup>''n''</sup> is [[homeomorphic]] to the sphere ''S''<sup>''m''+''n''</sup>.
			*The smash product of a space ''X'' with a circle is homeomorphic to the [[reduced suspension]] of ''X'':
			*:<math> \Sigma X \cong X \wedge S^1. \, </math>
			*The ''k''-fold iterated reduced suspension of ''X'' is homeomorphic to the smash product of ''X'' and a ''k''-sphere
			*:<math> \Sigma^k X \cong X \wedge S^k. \, </math>
			* In [[domain theory]], taking the product of two domains (so that the product is strict on its arguments).

			==As a symmetric monoidal product==
			For any pointed spaces ''X'', ''Y'', and ''Z'' in an appropriate "convenient" category (e.g. that of [[compactly generated space]]s) there are natural (basepoint preserving) [[homeomorphism]]s
			:<math>\begin{align}
			X \wedge Y &\cong Y\wedge X, \\
			(X\wedge Y)\wedge Z &\cong X \wedge (Y\wedge Z).
			\end{align}</math>
			However, for the naive category of pointed spaces, this fails. See the following discussion on MathOverflow.<ref>Omar Antolín-Camarena (mathoverflow.net/users/644), In which situations can one see that topological spaces are ill-behaved from the homotopical viewpoint?, http://mathoverflow.net/questions/76594 (version: 2011-09-28)</ref>

			These isomorphisms make the appropriate [[category of pointed spaces]] into a [[symmetric monoidal category]] with the smash product as the monoidal product and the pointed [[0-sphere]] (a two-point discrete space) as the unit object. One can therefore think of the smash product as a kind of [[tensor product]] in an appropriate category of pointed spaces.

			==Adjoint relationship==
			[[Adjoint functors]] make the analogy between the tensor product and the smash product more precise. In the category of [[module (mathematics)\|''R''-modules]] over a [[commutative ring]] ''R'', the tensor functor (– ⊗<sub>''R''</sub> ''A'') is left adjoint to the internal [[Hom functor]] Hom(''A'',–) so that:
			:<math>\mathrm{Hom}(X\otimes A,Y) \cong \mathrm{Hom}(X,\mathrm{Hom}(A,Y)).</math>
			In the [[category of pointed spaces]], the smash product plays the role of the tensor product. In particular, if ''A'' is [[locally compact Hausdorff]] then we have an adjunction
			:<math>\mathrm{Hom}(X\wedge A,Y) \cong \mathrm{Hom}(X,\mathrm{Hom}(A,Y))</math>
			where Hom(''A'',''Y'') is the space of based continuous maps together with the [[compact-open topology]].

			In particular, taking ''A'' to be the [[unit circle]] ''S''<sup>1</sup>, we see that the suspension functor Σ is left adjoint to the [[loop space]] functor Ω.
			:<math>\mathrm{Hom}(\Sigma X,Y) \cong \mathrm{Hom}(X,\Omega Y).</math>

			==References==
			{{Reflist}}
			*{{Hatcher AT}}

			{{DEFAULTSORT:Smash Product}}
			[[Category:Topology]]
			[[Category:Homotopy theory]]
			[[Category:Binary operations]]

en>Kasirbot: Robot: Adding fa:رده:الگوهای تبدیل سریع فوریه

2012-01-01T16:01:27Z

Robot: Adding fa:رده:الگوهای تبدیل سریع فوریه

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Gabrielle is what her life partner loves to call your sweetheart though she doesn't very like being called like that. Fish dealing with acne is something her partner doesn't really like nevertheless , she does. [http://managing.org/ Managing] people happens to be what she does although she plans on diaper changing it. For years she's been that reside in Massachusetts. Go to her own website to find out more: http://circuspartypanama.com<br><br>Here is my blog post hack clash of clans ([http://circuspartypanama.com linked internet site])

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en>Chobot: Bot: Migrating 1interwiki links, now provided by Wikidata on Q8217879

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