Gas engine - Revision history

en>RaymondShill: /* Tangye */

2014-12-26T11:18:31Z

Tangye

← Older revision		Revision as of 13:18, 26 December 2014
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	~~The author~~ is ~~known as Irwin~~ Wunder ~~but it's not the most masucline title out there~~. ~~To do aerobics~~ is ~~a factor that I'm totally addicted~~ to. ~~Years ago we moved to North Dakota~~. ~~My working day job~~ is a ~~meter reader~~.<br><br>~~Look at~~ my ~~website~~: [http://~~jewelrycase~~.co.kr/xe/~~Ring~~/~~11593 http:~~/~~/jewelrycase.co.kr~~]		Hello and welcome. My name is Numbers Wunder. North Dakota is her birth place but she will have to transfer 1 working day or an additional. Supervising is my profession. What I adore doing is doing ceramics but I haven't made a dime with it.<br><br>Feel free to surf to my webpage :: std home test - [http://faculty.jonahmancini.com/groups/solid-advice-for-dealing-with-a-candida-albicans/members/ mouse click the up coming internet site],

76.121.80.169: /* Thermal efficiency */ mispelling (an to a)

2014-02-05T18:48:09Z

Thermal efficiency: mispelling (an to a)

← Older revision		Revision as of 20:48, 5 February 2014
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	~~In [[mathematics]], a~~ '~~''filtered algebra''' is a generalization of~~ the ~~notion of a [[graded algebra]]~~. ~~Examples appear in many branches of [[mathematics]], especially in [[homological algebra]] and [[representation theory]].~~		The author is known as Irwin Wunder but it's not the most masucline title out there. To do aerobics is a factor that I'm totally addicted to. Years ago we moved to North Dakota. My working day job is a meter reader.<br><br>Look at my website: [http://jewelrycase.co.kr/xe/Ring/11593 http://jewelrycase.co.kr]

	~~A filtered algebra over the [[field (mathematics)\|field]] <math>k</math>~~ is ~~an [[Algebra over~~ a field\|algebra]] <math>(A,\cdot)</math> over <math>k</math> which has an increasing sequence <math> \{0\} \subset F_0 \subset F_1 \subset \cdots \subset F_i \subset \cdots \subset A </math> of subspaces of <math>A</math> such that

	~~:<math>A=\cup_{i\in \mathbb{N}} F_i</math>~~

	~~and that is compatible with the multiplication in the following sense~~

	~~:<math> \forall m,n \in \mathbb{N},\qquad F_m\cdot F_n\subset F_{n+m}.</math>~~

	~~==Associated graded algebra==~~
	~~In general there is the following construction that produces a graded algebra out of a filtered algebra.~~

	~~If <math>A</math> is a filtered algebra then the ''associated graded algebra'~~' ~~<math>\mathcal{G}(A)</math> is defined as follows: {{unordered list~~
	~~\|1= As a [[linear space\|vector space]]~~

	~~:<math> \mathcal{G}(A)=\bigoplus_{n\in \mathbb{N}}G_n\,, </math>~~

	~~where,~~

	~~:<math> G_0=F_0,</math> and~~

	~~:<math> \forall n>0, \quad G_n=F_n/F_{n-1}\,,</math>~~

	~~\|2= the multiplication is defined by~~

	~~:<math> (x+F_{n-1})(y+F_{~~m~~-1})=x\cdot y+F_{n+m-1}</math>~~

	~~for all <math>x\in F_n</math> and <math>y\in F_m</math>~~. ~~(More precisely, the multiplication map <math> \mathcal{G}(A)\times \mathcal{G}(A) \to \mathcal{G}(A)</math> is combined from the maps~~

	~~:<math> (F_n / F_{n-1}) \times (F_m / F_{m-1}) \~~to ~~F_{n+m}/F_{n+m-1}, \ \ \ \ \ \left(x+F_{n-1},y+F_{m-1}\right) \mapsto x\cdot y+F_{n+m-1}</math>~~

	~~for all <math>n\geq 0</math> and <math>m\geq 0</math>~~.)
	}}
	~~The multiplication~~ is ~~well defined and endows <math>\mathcal{G}(A)</math> with the structure of~~ a ~~graded algebra, with gradation <math>\{G_n\}_{n \in \mathbb{N}}~~.<~~/math~~> ~~Furthermore if~~ <~~math~~>~~A</math> is~~ [~~[associative]] then so is <math>\mathcal{G}(A)<~~/~~math>. Also if <math>A<~~/~~math> is unital, such that the unit lies in <math>F_0</math>, then <math>\mathcal{G}(A)</math> will be unital as well.~~

	As algebras <math>A</math> and <math>\mathcal{G}(A)</math> are distinct (with the exception of the trivial case that <math>A</math> is graded) but as vector spaces they are [[isomorphism\|isomorphic]].

	~~==Examples==~~
	~~Any [[graded algebra]] graded by ℕ, for example <math>A = \oplus_{n\in \mathbb{N}} A_n </math>, has a filtration given by <math> F_n = \oplus_{i=0}^n A_i </math>~~.

	~~An example of a filtered algebra is the [[Clifford algebra]] <math>\mathrm{Cliff}(V,q)<~~/~~math> of a vector space <math>V</math> endowed with a [[polarization identity\|quadratic form]] <math>q.<~~/~~math> The associated graded algebra is <math>\bigwedge V<~~/~~math>, the [[exterior algebra]] of <math>V.</math>~~

	~~The [[symmetric algebra]] on the dual of an [[affine space]] is a filtered algebra of polynomials; on a [[vector space]], one instead obtains a graded algebra.~~

	The [[universal enveloping algebra]] of a [[Lie algebra]] <math>\mathfrak{g}</math> is also naturally filtered. The [[PBW theorem]] states that the associated graded algebra is simply <math>\mathrm{Sym} (\mathfrak{g})</math>.

	Scalar [[differential operator]]s on a manifold <math>M</math> form a filtered algebra where the filtration is given by the degree of differential operators. The associated graded is the commutative algebra of smooth functions on the cotangent bundle <math>T^M</math> which are polynomial along the fibers of the projection <math>\pi\colon T^M\rightarrow M</math>.

	~~The [[group ring\|group algebra]] of a group with a [[length function]] is a filtered algebra.~~

	~~==See also==~~
	* [[Filtration (mathematics)]]
	* [[Length function]]

	~~==References==~~
	*{{cite book\|last=Abe\|first=Eiichi\|title=Hopf Algebras\|year=1980\|publisher=Cambridge University Press\|location=Cambridge\|isbn=0-521-22240-0\|url=http://~~books~~.~~google~~.ch/books?id=D0AIcewz5-8C&pg=PR4&lpg=PR4&dq=isbn+0-521-22240-0&source=bl&ots=hghAhwKyW6&sig=q9kGyfSJ3GeWFgO51uecxVCu7bo&hl=en&sa=X&ei=1AVNT7HjDtP64QTx08TyAg&ved=0CCUQ6AEwAQ#v=onepage&q=isbn%200-521-22240-0&f=false}}

	~~{{PlanetMath attribution\|id=3938\|title=Filtered algebra}}~~

	~~[[Category:Algebras]]~~
	~~[[Category:Homological algebra]~~]

en>Dewritech: /* History */clean up, typo(s) fixed: follwed → followed using AWB

2013-11-13T18:50:31Z

History: clean up, typo(s) fixed: follwed → followed using AWB

← Older revision		Revision as of 20:50, 13 November 2013
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	~~Greetings~~. ~~Let me begin by telling you~~ the ~~author~~'~~s name~~ - ~~Phebe~~. ~~Puerto Rico~~ is ~~where he's usually been residing but she requirements~~ to ~~move simply because~~ of ~~her family members~~. ~~What I love doing~~ is ~~playing baseball~~ but ~~I haven't made~~ a ~~dime~~ with it. ~~She~~ is a ~~librarian but she'~~s ~~usually wanted her own company~~.<br><br>~~Here~~ is ~~my web~~-~~site: std home test~~ - [http://~~Www~~.~~1A-Pornotube~~.~~com~~/~~user/GCalvert have a peek at these guys~~],		In [[mathematics]], a '''filtered algebra''' is a generalization of the notion of a [[graded algebra]]. Examples appear in many branches of [[mathematics]], especially in [[homological algebra]] and [[representation theory]].

			A filtered algebra over the [[field (mathematics)\|field]] <math>k</math> is an [[Algebra over a field\|algebra]] <math>(A,\cdot)</math> over <math>k</math> which has an increasing sequence <math> \{0\} \subset F_0 \subset F_1 \subset \cdots \subset F_i \subset \cdots \subset A </math> of subspaces of <math>A</math> such that

			:<math>A=\cup_{i\in \mathbb{N}} F_i</math>

			and that is compatible with the multiplication in the following sense

			:<math> \forall m,n \in \mathbb{N},\qquad F_m\cdot F_n\subset F_{n+m}.</math>

			==Associated graded algebra==
			In general there is the following construction that produces a graded algebra out of a filtered algebra.

			If <math>A</math> is a filtered algebra then the ''associated graded algebra'' <math>\mathcal{G}(A)</math> is defined as follows: {{unordered list
			\|1= As a [[linear space\|vector space]]

			:<math> \mathcal{G}(A)=\bigoplus_{n\in \mathbb{N}}G_n\,, </math>

			where,

			:<math> G_0=F_0,</math> and

			:<math> \forall n>0, \quad G_n=F_n/F_{n-1}\,,</math>

			\|2= the multiplication is defined by

			:<math> (x+F_{n-1})(y+F_{m-1})=x\cdot y+F_{n+m-1}</math>

			for all <math>x\in F_n</math> and <math>y\in F_m</math>. (More precisely, the multiplication map <math> \mathcal{G}(A)\times \mathcal{G}(A) \to \mathcal{G}(A)</math> is combined from the maps

			:<math> (F_n / F_{n-1}) \times (F_m / F_{m-1}) \to F_{n+m}/F_{n+m-1}, \ \ \ \ \ \left(x+F_{n-1},y+F_{m-1}\right) \mapsto x\cdot y+F_{n+m-1}</math>

			for all <math>n\geq 0</math> and <math>m\geq 0</math>.)
			}}
			The multiplication is well defined and endows <math>\mathcal{G}(A)</math> with the structure of a graded algebra, with gradation <math>\{G_n\}_{n \in \mathbb{N}}.</math> Furthermore if <math>A</math> is [[associative]] then so is <math>\mathcal{G}(A)</math>. Also if <math>A</math> is unital, such that the unit lies in <math>F_0</math>, then <math>\mathcal{G}(A)</math> will be unital as well.

			As algebras <math>A</math> and <math>\mathcal{G}(A)</math> are distinct (with the exception of the trivial case that <math>A</math> is graded) but as vector spaces they are [[isomorphism\|isomorphic]].

			==Examples==
			Any [[graded algebra]] graded by ℕ, for example <math>A = \oplus_{n\in \mathbb{N}} A_n </math>, has a filtration given by <math> F_n = \oplus_{i=0}^n A_i </math>.

			An example of a filtered algebra is the [[Clifford algebra]] <math>\mathrm{Cliff}(V,q)</math> of a vector space <math>V</math> endowed with a [[polarization identity\|quadratic form]] <math>q.</math> The associated graded algebra is <math>\bigwedge V</math>, the [[exterior algebra]] of <math>V.</math>

			The [[symmetric algebra]] on the dual of an [[affine space]] is a filtered algebra of polynomials; on a [[vector space]], one instead obtains a graded algebra.

			The [[universal enveloping algebra]] of a [[Lie algebra]] <math>\mathfrak{g}</math> is also naturally filtered. The [[PBW theorem]] states that the associated graded algebra is simply <math>\mathrm{Sym} (\mathfrak{g})</math>.

			Scalar [[differential operator]]s on a manifold <math>M</math> form a filtered algebra where the filtration is given by the degree of differential operators. The associated graded is the commutative algebra of smooth functions on the cotangent bundle <math>T^M</math> which are polynomial along the fibers of the projection <math>\pi\colon T^M\rightarrow M</math>.

			The [[group ring\|group algebra]] of a group with a [[length function]] is a filtered algebra.

			==See also==
			* [[Filtration (mathematics)]]
			* [[Length function]]

			==References==
			*{{cite book\|last=Abe\|first=Eiichi\|title=Hopf Algebras\|year=1980\|publisher=Cambridge University Press\|location=Cambridge\|isbn=0-521-22240-0\|url=http://books.google.ch/books?id=D0AIcewz5-8C&pg=PR4&lpg=PR4&dq=isbn+0-521-22240-0&source=bl&ots=hghAhwKyW6&sig=q9kGyfSJ3GeWFgO51uecxVCu7bo&hl=en&sa=X&ei=1AVNT7HjDtP64QTx08TyAg&ved=0CCUQ6AEwAQ#v=onepage&q=isbn%200-521-22240-0&f=false}}

			{{PlanetMath attribution\|id=3938\|title=Filtered algebra}}

			[[Category:Algebras]]
			[[Category:Homological algebra]]

en>Biscuittin: link

2012-06-11T11:40:20Z

link

New page

Greetings. Let me begin by telling you the author's name - Phebe. Puerto Rico is where he's usually been residing but she requirements to move simply because of her family members. What I love doing is playing baseball but I haven't made a dime with it. She is a librarian but she's usually wanted her own company.<br><br>Here is my web-site: std home test - [http://Www.1A-Pornotube.com/user/GCalvert have a peek at these guys],