en>Yobot: WP:CHECKWIKI error fixes + other fixes using AWB (10065)

2014-03-29T08:43:34Z

WP:CHECKWIKI error fixes + other fixes using AWB (10065)

← Older revision		Revision as of 10:43, 29 March 2014
Line 1:		Line 1:
	~~In [[mathematics]], an '''analytic semigroup'''~~ is ~~particular kind of [[C0 semigroup\|strongly continuous semigroup]]. Analytic semigroups are used in~~ the solution of [[partial differential equations]]; compared to strongly continuous semigroups, analytic semigroups provide better [[smooth function\|regularity]] of solutions to initial value problems, better results concerning perturbations of the [[C0 semigroup#Infinitesimal generator\|infinitesimal generator]], and a relationship between the type of the semigroup and the [[spectrum (functional analysis)\|spectrum]] of the infinitesimal generator.		Alyson Meagher is the title her parents gave her but she doesn't like when people use her complete title. Kentucky is exactly where I've always been living. My husband doesn't like it the way I do but what I truly like performing is caving but I don't have the time lately. My day job is an info live psychic reading ([http://www.chk.woobi.co.kr/xe/?document_srl=346069 http://www.chk.woobi.co.kr/xe/?document_srl=346069]) officer but I've already [http://m-card.co.kr/xe/mcard_2013_promote01/29877 free psychic readings] applied for an additional 1.<br><br>My web site ... [http://www.january-yjm.com/xe/index.php?mid=video&document_srl=158289 best psychics]

	~~==Definition==~~

	~~Let Γ(~~''t'~~') = exp(''At'') be a strongly continuous one-parameter semigroup on a [[Banach space]] (''X'', \|\|·\|\|) with infinitesimal generator ''A''~~. Γ is ~~said to be~~ an ~~'''analytic semigroup''' if~~

	* for some 0 < ''θ'' < ''π'' ⁄ 2, the [~~[continuous linear operator]] exp(''At'') ~~:~~ ''X'' → ''X'' can be extended to ''t'' ∈ Δ<sub>''θ''<~~/~~sub>,~~

	~~::<math>\Delta_{\theta}~~ = ~~\{ 0 \} \cup \{ t \in \mathbb{C}~~ : ~~\| \mathrm{arg}(t) \| < \theta \},<~~/~~math>~~

	~~:and the usual semigroup conditions hold for ''s'', ''t'' ∈ Δ<sub>''θ''<~~/sub>: exp(''A''0) = id, exp(''A''(''t'' + ''s'')) = exp(''At'')exp(''As''), and, for each ''x'' ∈ ''X'', exp(''At'')''x'' is [[continuous function\|continuous]] in ''t'';

	* and, for all ''t'' ∈ Δ<sub>''θ''</~~sub> \ {0}, exp(''At'') is [[analytic function\|analytic]] in ''t'' in the sense of the [[uniform topology\|uniform operator topology]].~~

	=~~=Characterization==~~

	~~The infinitesimal generators of analytic semigroups have the following characterization:~~

	~~A [[closed operator\|closed]], [[dense set\|densely-defined]] [[linear operator]~~] ~~''A'' on a Banach space ''X'' is the generator of an analytic semigroup [[if and only if]] there exists an ''ω'' ∈ '''R''' such that the [[half-plane]] Re(''λ''~~)~~ > ~~'~~'ω'' is contained in the~~ [~~[resolvent formalism\|resolvent set]] of ''A'' and, moreover, there is a constant ''C'' such that~~

	:~~<math>\\| R_{\lambda} (A) \\| \leq \frac{C}{\| \lambda~~ - ~~\omega \|}<~~/~~math>~~

	~~for Re(''λ'') > ''ω'' and where <math>R_\lambda(A)<~~/~~math> is the [[Resolvent_formalism\|resolvent~~]~~] of the operator ''A''~~. ~~If this is the case, then the resolvent set actually contains a sector of the form~~

	:<~~math~~>~~\left\{ \lambda \in \mathbf{C} : \| \mathrm{arg} (\lambda - \omega) \| < \frac{\pi}{2} + \delta \right\}~~<~~/math~~>

	~~for some ''δ'' > 0, and an analogous resolvent estimate holds in this sector~~. ~~Moreover, the semigroup is represented by~~

	:~~<math>\exp (At) = \frac1{2 \pi i} \int_{\gamma} e^{\lambda t} ( \lambda \mathrm{id} - A )^{-1} \, \mathrm{d} \lambda,<~~/~~math>~~

	~~where ''γ'' is any curve from ''e''<sup>−''iθ''<~~/~~sup>∞ to ''e''<sup>+''iθ''<~~/~~sup>∞ such that ''γ'' lies entirely in the sector~~

	~~:<math>\big\{ \lambda \in \mathbf{C} : \| \mathrm{arg} (\lambda - \omega) \| \leq \theta \big\},<~~/~~math>~~

	~~with ''π'' ⁄ 2 < ''θ'' < ''π'' ⁄ 2 + ''δ''~~.

	==~~References==~~

	* {{cite book
	~~\| last = Renardy~~
	~~\| first = Michael~~
	~~\| coauthors = Rogers, Robert C.~~
	~~\| title = An introduction to partial differential equations~~
	~~\| series = Texts in Applied Mathematics 13~~
	~~\| edition = Second edition~~
	~~\| publisher = Springer-Verlag~~
	~~\| location = New York~~
	~~\| year = 2004~~
	~~\| pages = xiv+434~~
	~~\| isbn = 0-387-00444-0~~
	~~\| mr = 2028503~~
	}}

	~~[[Category:Functional analysis]]~~
	~~[[Category:Partial differential equations]]~~
	~~[[Category:Semigroup theory]~~]

en>Cydebot: Robot - Moving category Space exploration to Spaceflight per CFD at Wikipedia:Categories for discussion/Log/2011 January 10.

2011-01-18T20:07:00Z

Robot - Moving category Space exploration to Spaceflight per CFD at Wikipedia:Categories for discussion/Log/2011 January 10.

New page

In [[mathematics]], an '''analytic semigroup''' is particular kind of [[C0 semigroup|strongly continuous semigroup]]. Analytic semigroups are used in the solution of [[partial differential equations]]; compared to strongly continuous semigroups, analytic semigroups provide better [[smooth function|regularity]] of solutions to initial value problems, better results concerning perturbations of the [[C0 semigroup#Infinitesimal generator|infinitesimal generator]], and a relationship between the type of the semigroup and the [[spectrum (functional analysis)|spectrum]] of the infinitesimal generator.

==Definition==

Let Γ(''t'') = exp(''At'') be a strongly continuous one-parameter semigroup on a [[Banach space]] (''X'', ||·||) with infinitesimal generator ''A''. Γ is said to be an '''analytic semigroup''' if

* for some 0 < ''θ'' < ''π'' ⁄ 2, the [[continuous linear operator]] exp(''At'') : ''X'' → ''X'' can be extended to ''t'' ∈ Δ''θ'',

::<math>\Delta_{\theta} = \{ 0 \} \cup \{ t \in \mathbb{C} : | \mathrm{arg}(t) | < \theta \},</math>

:and the usual semigroup conditions hold for ''s'', ''t'' ∈ Δ''θ'': exp(''A''0) = id, exp(''A''(''t'' + ''s'')) = exp(''At'')exp(''As''), and, for each ''x'' ∈ ''X'', exp(''At'')''x'' is [[continuous function|continuous]] in ''t'';

* and, for all ''t'' ∈ Δ''θ'' \ {0}, exp(''At'') is [[analytic function|analytic]] in ''t'' in the sense of the [[uniform topology|uniform operator topology]].

==Characterization==

The infinitesimal generators of analytic semigroups have the following characterization:

A [[closed operator|closed]], [[dense set|densely-defined]] [[linear operator]] ''A'' on a Banach space ''X'' is the generator of an analytic semigroup [[if and only if]] there exists an ''ω'' ∈ '''R''' such that the [[half-plane]] Re(''λ'') > ''ω'' is contained in the [[resolvent formalism|resolvent set]] of ''A'' and, moreover, there is a constant ''C'' such that

:<math>\| R_{\lambda} (A) \| \leq \frac{C}{| \lambda - \omega |}</math>

for Re(''λ'') > ''ω'' and where <math>R_\lambda(A)</math> is the [[Resolvent_formalism|resolvent]] of the operator ''A''. If this is the case, then the resolvent set actually contains a sector of the form

:<math>\left\{ \lambda \in \mathbf{C} : | \mathrm{arg} (\lambda - \omega) | < \frac{\pi}{2} + \delta \right\}</math>

for some ''δ'' > 0, and an analogous resolvent estimate holds in this sector. Moreover, the semigroup is represented by

:<math>\exp (At) = \frac1{2 \pi i} \int_{\gamma} e^{\lambda t} ( \lambda \mathrm{id} - A )^{-1} \, \mathrm{d} \lambda,</math>

where ''γ'' is any curve from ''e''−''iθ''∞ to ''e''+''iθ''∞ such that ''γ'' lies entirely in the sector

:<math>\big\{ \lambda \in \mathbf{C} : | \mathrm{arg} (\lambda - \omega) | \leq \theta \big\},</math>

with ''π'' ⁄ 2 < ''θ'' < ''π'' ⁄ 2 + ''δ''.

==References==

* {{cite book
| last = Renardy
| first = Michael
| coauthors = Rogers, Robert C.
| title = An introduction to partial differential equations
| series = Texts in Applied Mathematics 13
| edition = Second edition
| publisher = Springer-Verlag
| location = New York
| year = 2004
| pages = xiv+434
| isbn = 0-387-00444-0
| mr = 2028503
}}

[[Category:Functional analysis]]
[[Category:Partial differential equations]]
[[Category:Semigroup theory]]

← Older revision		Revision as of 10:43, 29 March 2014
Line 1:		Line 1:
	~~In [[mathematics]], an '''analytic semigroup'''~~ is ~~particular kind of [[C0 semigroup\|strongly continuous semigroup]]. Analytic semigroups are used in~~ the solution of [[partial differential equations]]; compared to strongly continuous semigroups, analytic semigroups provide better [[smooth function\|regularity]] of solutions to initial value problems, better results concerning perturbations of the [[C0 semigroup#Infinitesimal generator\|infinitesimal generator]], and a relationship between the type of the semigroup and the [[spectrum (functional analysis)\|spectrum]] of the infinitesimal generator.		Alyson Meagher is the title her parents gave her but she doesn't like when people use her complete title. Kentucky is exactly where I've always been living. My husband doesn't like it the way I do but what I truly like performing is caving but I don't have the time lately. My day job is an info live psychic reading ([http://www.chk.woobi.co.kr/xe/?document_srl=346069 http://www.chk.woobi.co.kr/xe/?document_srl=346069]) officer but I've already [http://m-card.co.kr/xe/mcard_2013_promote01/29877 free psychic readings] applied for an additional 1.<br><br>My web site ... [http://www.january-yjm.com/xe/index.php?mid=video&document_srl=158289 best psychics]

	~~==Definition==~~

	~~Let Γ(~~''t'~~') = exp(''At'') be a strongly continuous one-parameter semigroup on a [[Banach space]] (''X'', \|\|·\|\|) with infinitesimal generator ''A''~~. Γ is ~~said to be~~ an ~~'''analytic semigroup''' if~~

	* for some 0 < ''θ'' < ''π'' ⁄ 2, the [~~[continuous linear operator]] exp(''At'') ~~:~~ ''X'' → ''X'' can be extended to ''t'' ∈ Δ<sub>''θ''<~~/~~sub>,~~

	~~::<math>\Delta_{\theta}~~ = ~~\{ 0 \} \cup \{ t \in \mathbb{C}~~ : ~~\| \mathrm{arg}(t) \| < \theta \},<~~/~~math>~~

	~~:and the usual semigroup conditions hold for ''s'', ''t'' ∈ Δ<sub>''θ''<~~/sub>: exp(''A''0) = id, exp(''A''(''t'' + ''s'')) = exp(''At'')exp(''As''), and, for each ''x'' ∈ ''X'', exp(''At'')''x'' is [[continuous function\|continuous]] in ''t'';

	* and, for all ''t'' ∈ Δ<sub>''θ''</~~sub> \ {0}, exp(''At'') is [[analytic function\|analytic]] in ''t'' in the sense of the [[uniform topology\|uniform operator topology]].~~

	=~~=Characterization==~~

	~~The infinitesimal generators of analytic semigroups have the following characterization:~~

	~~A [[closed operator\|closed]], [[dense set\|densely-defined]] [[linear operator]~~] ~~''A'' on a Banach space ''X'' is the generator of an analytic semigroup [[if and only if]] there exists an ''ω'' ∈ '''R''' such that the [[half-plane]] Re(''λ''~~)~~ > ~~'~~'ω'' is contained in the~~ [~~[resolvent formalism\|resolvent set]] of ''A'' and, moreover, there is a constant ''C'' such that~~

	:~~<math>\\| R_{\lambda} (A) \\| \leq \frac{C}{\| \lambda~~ - ~~\omega \|}<~~/~~math>~~

	~~for Re(''λ'') > ''ω'' and where <math>R_\lambda(A)<~~/~~math> is the [[Resolvent_formalism\|resolvent~~]~~] of the operator ''A''~~. ~~If this is the case, then the resolvent set actually contains a sector of the form~~

	:<~~math~~>~~\left\{ \lambda \in \mathbf{C} : \| \mathrm{arg} (\lambda - \omega) \| < \frac{\pi}{2} + \delta \right\}~~<~~/math~~>

	~~for some ''δ'' > 0, and an analogous resolvent estimate holds in this sector~~. ~~Moreover, the semigroup is represented by~~

	:~~<math>\exp (At) = \frac1{2 \pi i} \int_{\gamma} e^{\lambda t} ( \lambda \mathrm{id} - A )^{-1} \, \mathrm{d} \lambda,<~~/~~math>~~

	~~where ''γ'' is any curve from ''e''<sup>−''iθ''<~~/~~sup>∞ to ''e''<sup>+''iθ''<~~/~~sup>∞ such that ''γ'' lies entirely in the sector~~

	~~:<math>\big\{ \lambda \in \mathbf{C} : \| \mathrm{arg} (\lambda - \omega) \| \leq \theta \big\},<~~/~~math>~~

	~~with ''π'' ⁄ 2 < ''θ'' < ''π'' ⁄ 2 + ''δ''~~.

	==~~References==~~

	* {{cite book
	~~\| last = Renardy~~
	~~\| first = Michael~~
	~~\| coauthors = Rogers, Robert C.~~
	~~\| title = An introduction to partial differential equations~~
	~~\| series = Texts in Applied Mathematics 13~~
	~~\| edition = Second edition~~
	~~\| publisher = Springer-Verlag~~
	~~\| location = New York~~
	~~\| year = 2004~~
	~~\| pages = xiv+434~~
	~~\| isbn = 0-387-00444-0~~
	~~\| mr = 2028503~~
	}}

	~~[[Category:Functional analysis]]~~
	~~[[Category:Partial differential equations]]~~
	~~[[Category:Semigroup theory]~~]

Gate orbit - Revision history

en>Yobot: WP:CHECKWIKI error fixes + other fixes using AWB (10065)

en>Cydebot: Robot - Moving category Space exploration to Spaceflight per CFD at Wikipedia:Categories for discussion/Log/2011 January 10.