Engine efficiency - Revision history

71.145.204.236: /* Compression ratio */ expansion ratio is the most important factor for engine efficiency . The Otto / Langen engine was a non compression engine but had a very good thermal efficiency due to the large expansion ratio. http://en.wikipedia.org/wiki/Nik

2014-12-16T15:31:52Z

Compression ratio: expansion ratio is the most important factor for engine efficiency . The Otto / Langen engine was a non compression engine but had a very good thermal efficiency due to the large expansion ratio. http://en.wikipedia.org/wiki/Nik

← Older revision		Revision as of 17:31, 16 December 2014
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	~~She is known by~~ the title ~~of Myrtle Shryock. One of the very best issues in the world for him~~ is ~~to gather badges~~ but ~~he is struggling to discover time for~~ it. ~~Her family members lives in Minnesota~~. In her ~~expert life~~ she is ~~a payroll clerk but~~ she's always ~~needed~~ her ~~personal company~~.<br><br>~~My site~~ [http://~~change-university~~.~~com/groups~~/~~curing-your-yeast-infection-how-to-accomplish-it-easily~~/ home std ~~test kit~~]		Hello. std testing at home Allow me introduce the author. Her title is Emilia Shroyer but it's not [http://www.revleft.com/vb/member.php?u=160656 over the counter std test] most female title out there. Hiring is her working day job now and she will not alter it whenever soon. The factor she adores most is [http://www.onhealth.com/genital_herpes_in_women/article.htm std testing] [http://nfldev.com/index.php?do=/profile-12302/info/ at home std test] home body building and home std test kit now she is trying to earn cash with it. Puerto Rico is where he's always been living but she requirements to transfer simply because of her family.<br><br>Also visit my webpage: [http://www.pornextras.info/blog/129353 at home std testing]

en>Epicgenius: Reverted 1 edit by 117.239.47.178 identified as test/vandalism using STiki (Mistake? Report it.)

2014-02-24T14:22:29Z

Reverted 1 edit by 117.239.47.178 identified as test/vandalism using STiki (Mistake? Report it.)

← Older revision		Revision as of 16:22, 24 February 2014
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	~~In [[mathematics]], '''Ehrling's lemma'''~~ is ~~a result concerning [[Banach space]]s. It is often used in [[functional analysis]] to demonstrate~~ the ~~[[norm (mathematics)#Properties\|equivalence]]~~ of ~~certain [[norm (mathematics)\|norms]] on [[Sobolev space]]s~~.		She is known by the title of Myrtle Shryock. One of the very best issues in the world for him is to gather badges but he is struggling to discover time for it. Her family members lives in Minnesota. In her expert life she is a payroll clerk but she's always needed her personal company.<br><br>My site [http://change-university.com/groups/curing-your-yeast-infection-how-to-accomplish-it-easily/ home std test kit]

	~~==Statement~~ of the ~~lemma==~~

	~~Let (''X'', \|\|·\|\|<sub>''X''</sub>), (''Y'', \|\|·\|\|<sub>''Y''</sub>) and (''Z'', \|\|·\|\|<sub>''Z''</sub>) be three Banach spaces. Assume that:~~
	* ''X'' is [[compactly embedded]] in ~~''Y'': i.e. ''X'' &sube; ''Y'' and every \|\|·\|\|<sub>''X''</sub>-[[bounded function\|bounded]] [[sequence]] in ''X'' has a [[subsequence]] that~~ is ~~\|\|·\|\|<sub>''Y''</sub>-[[Limit (mathematics)\|convergent]]; and~~
	* ''Y'' is ~~[[continuously embedded]]~~ in ~~''Z'': i~~.~~e. ''Y'' &sube; ''Z'' and there~~ is a ~~constant ''k'' so that \|\|''y''\|\|<sub>''Z''</sub> ≤ ''k''\|\|''y''\|\|<sub>''Y''</sub> for every ''y'' ∈ ''Y'~~'.
	~~Then, for every ''ε'' > 0, there exists a constant ''C''(''ε'') such that, for all ''x'' ∈ ''X'',~~

	~~:<math>\\| x \\|_{Y} \leq \varepsilon \\| x \\|_{X} + C(\varepsilon) \\| x \\|_{Z}</math>~~

	~~==Corollary (equivalent norms for Sobolev spaces)==~~

	~~Let Ω ⊂ '''R'''~~<~~sup~~>~~''n''~~<~~/sup~~> be [~~[open set\|open]] and [[bounded set\|bounded]], and let ''k'' ∈ '''N'''. Suppose that the Sobolev space ''H''<sup>''k''<~~/~~sup>(Ω) is compactly embedded in ''H''<sup>''k''−1<~~/~~sup>(Ω)~~. ~~Then the following two norms on ''H''<sup>''k''<~~/~~sup>(Ω) are equivalent:~~

	~~:<math>\\| \cdot \\| : H^{k} (\Omega) \to \mathbf{R}: u \mapsto \\| u \\| := \sqrt{\sum_{\| \alpha \| \leq k} \\| \mathrm{D}^{\alpha} u \\|_{L^{2} (\Omega)}^{2}}<~~/~~math>~~

	~~and~~

	~~:<math>\\| \cdot \\|' : H^{k} (\Omega) \to \mathbf{R}: u \mapsto \\| u \\|' := \sqrt{\\| u \\|_{L^{1} (\Omega)}^{2} + \sum_{\| \alpha \| = k} \\| \mathrm{D}^{\alpha} u \\|_{L^{2} (\Omega)}^{2}}.</math>~~

	For the subspace of ''H''<sup>''k''</sup>(Ω) consisting of those Sobolev functions with [[trace operator\|zero trace]] (those that are "zero on the boundary" of Ω), the ''L''<sup>1</sup> norm of ''u'' can be left out to yield another equivalent norm.

	~~==References==~~

	* {{cite book
	~~\| last1 = Renardy~~
	~~\| first1 = Michael~~
	~~\| last2 = Rogers~~
	~~\| first2 = Robert C.~~
	~~\| title = An Introduction to Partial Differential Equations~~
	~~\| publisher = Springer~~-~~Verlag~~
	~~\| location = Berlin~~
	~~\| year=1992~~
	~~\| isbn=978~~-3-~~540~~-~~97952~~-4
	}}

	~~[[Category:Banach spaces~~]]
	~~[[Category:Sobolev spaces]]~~
	~~[[Category:Lemmas]]~~

	~~{{mathanalysis-stub}}~~

en>APerson: Reverted 1 good faith edit by 115.251.246.210 using STiki

2014-02-02T04:15:00Z

Reverted 1 good faith edit by 115.251.246.210 using STiki

← Older revision		Revision as of 06:15, 2 February 2014
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	~~Hello and welcome~~. ~~at home std testing My title~~ is ~~Irwin and~~ [~~http://www~~.~~onlinedatingmagazine.com~~/~~STDs~~/~~STDadvice~~/~~treatablestds~~.~~html I completely] dig~~ that ~~title. Puerto Rico~~ is ~~exactly where he~~'~~s always been living [http~~:~~//Www~~.~~onlinedatingmagazine~~.~~com~~/~~STDs~~/~~STDadvice/treatablestds.html std testing~~] ~~at home but she requirements to transfer simply because of her family~~ [~~http~~:~~//javly~~.~~com/?p=56362 std testing at home] members~~. ~~Supervising~~ is ~~my occupation. His spouse doesn~~'~~t like it [http:~~//~~www~~.~~adosphere.com~~/~~member~~/~~28512/blog/view/66112/ over the counter std test~~] ~~way he does but what he truly likes performing is to do aerobics~~ and he'~~s been doing it for quite a whilst~~.<br><br>~~My page - [http:~~/~~/www~~.~~reachoutsociety.info~~/~~user~~/~~JCondon http~~://~~www~~.~~reachoutsociety~~.~~info/~~]		In [[mathematics]], '''Ehrling's lemma''' is a result concerning [[Banach space]]s. It is often used in [[functional analysis]] to demonstrate the [[norm (mathematics)#Properties\|equivalence]] of certain [[norm (mathematics)\|norms]] on [[Sobolev space]]s.

			==Statement of the lemma==

			Let (''X'', \|\|·\|\|<sub>''X''</sub>), (''Y'', \|\|·\|\|<sub>''Y''</sub>) and (''Z'', \|\|·\|\|<sub>''Z''</sub>) be three Banach spaces. Assume that:
			* ''X'' is [[compactly embedded]] in ''Y'': i.e. ''X'' &sube; ''Y'' and every \|\|·\|\|<sub>''X''</sub>-[[bounded function\|bounded]] [[sequence]] in ''X'' has a [[subsequence]] that is \|\|·\|\|<sub>''Y''</sub>-[[Limit (mathematics)\|convergent]]; and
			* ''Y'' is [[continuously embedded]] in ''Z'': i.e. ''Y'' &sube; ''Z'' and there is a constant ''k'' so that \|\|''y''\|\|<sub>''Z''</sub> ≤ ''k''\|\|''y''\|\|<sub>''Y''</sub> for every ''y'' ∈ ''Y''.
			Then, for every ''ε'' > 0, there exists a constant ''C''(''ε'') such that, for all ''x'' ∈ ''X'',

			:<math>\\| x \\|_{Y} \leq \varepsilon \\| x \\|_{X} + C(\varepsilon) \\| x \\|_{Z}</math>

			==Corollary (equivalent norms for Sobolev spaces)==

			Let Ω ⊂ '''R'''<sup>''n''</sup> be [[open set\|open]] and [[bounded set\|bounded]], and let ''k'' ∈ '''N'''. Suppose that the Sobolev space ''H''<sup>''k''</sup>(Ω) is compactly embedded in ''H''<sup>''k''−1</sup>(Ω). Then the following two norms on ''H''<sup>''k''</sup>(Ω) are equivalent:

			:<math>\\| \cdot \\| : H^{k} (\Omega) \to \mathbf{R}: u \mapsto \\| u \\| := \sqrt{\sum_{\| \alpha \| \leq k} \\| \mathrm{D}^{\alpha} u \\|_{L^{2} (\Omega)}^{2}}</math>

			and

			:<math>\\| \cdot \\|' : H^{k} (\Omega) \to \mathbf{R}: u \mapsto \\| u \\|' := \sqrt{\\| u \\|_{L^{1} (\Omega)}^{2} + \sum_{\| \alpha \| = k} \\| \mathrm{D}^{\alpha} u \\|_{L^{2} (\Omega)}^{2}}.</math>

			For the subspace of ''H''<sup>''k''</sup>(Ω) consisting of those Sobolev functions with [[trace operator\|zero trace]] (those that are "zero on the boundary" of Ω), the ''L''<sup>1</sup> norm of ''u'' can be left out to yield another equivalent norm.

			==References==

			* {{cite book
			\| last1 = Renardy
			\| first1 = Michael
			\| last2 = Rogers
			\| first2 = Robert C.
			\| title = An Introduction to Partial Differential Equations
			\| publisher = Springer-Verlag
			\| location = Berlin
			\| year=1992
			\| isbn=978-3-540-97952-4
			}}

			[[Category:Banach spaces]]
			[[Category:Sobolev spaces]]
			[[Category:Lemmas]]

			{{mathanalysis-stub}}

en>Mink Butler Davenport: /* Gasoline (petrol) Engines */ comparative symmetry & uniformity

2012-08-31T17:25:46Z

Gasoline (petrol) Engines: comparative symmetry & uniformity

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← Older revision		Revision as of 16:22, 24 February 2014
Line 1:		Line 1:
	~~In [[mathematics]], '''Ehrling's lemma'''~~ is ~~a result concerning [[Banach space]]s. It is often used in [[functional analysis]] to demonstrate~~ the ~~[[norm (mathematics)#Properties\|equivalence]]~~ of ~~certain [[norm (mathematics)\|norms]] on [[Sobolev space]]s~~.		She is known by the title of Myrtle Shryock. One of the very best issues in the world for him is to gather badges but he is struggling to discover time for it. Her family members lives in Minnesota. In her expert life she is a payroll clerk but she's always needed her personal company.<br><br>My site [http://change-university.com/groups/curing-your-yeast-infection-how-to-accomplish-it-easily/ home std test kit]

	~~==Statement~~ of the ~~lemma==~~

	~~Let (''X'', \|\|·\|\|<sub>''X''</sub>), (''Y'', \|\|·\|\|<sub>''Y''</sub>) and (''Z'', \|\|·\|\|<sub>''Z''</sub>) be three Banach spaces. Assume that:~~
	* ''X'' is [[compactly embedded]] in ~~''Y'': i.e. ''X'' &sube; ''Y'' and every \|\|·\|\|<sub>''X''</sub>-[[bounded function\|bounded]] [[sequence]] in ''X'' has a [[subsequence]] that~~ is ~~\|\|·\|\|<sub>''Y''</sub>-[[Limit (mathematics)\|convergent]]; and~~
	* ''Y'' is ~~[[continuously embedded]]~~ in ~~''Z'': i~~.~~e. ''Y'' &sube; ''Z'' and there~~ is a ~~constant ''k'' so that \|\|''y''\|\|<sub>''Z''</sub> ≤ ''k''\|\|''y''\|\|<sub>''Y''</sub> for every ''y'' ∈ ''Y'~~'.
	~~Then, for every ''ε'' > 0, there exists a constant ''C''(''ε'') such that, for all ''x'' ∈ ''X'',~~

	~~:<math>\\| x \\|_{Y} \leq \varepsilon \\| x \\|_{X} + C(\varepsilon) \\| x \\|_{Z}</math>~~

	~~==Corollary (equivalent norms for Sobolev spaces)==~~

	~~Let Ω ⊂ '''R'''~~<~~sup~~>~~''n''~~<~~/sup~~> be [~~[open set\|open]] and [[bounded set\|bounded]], and let ''k'' ∈ '''N'''. Suppose that the Sobolev space ''H''<sup>''k''<~~/~~sup>(Ω) is compactly embedded in ''H''<sup>''k''−1<~~/~~sup>(Ω)~~. ~~Then the following two norms on ''H''<sup>''k''<~~/~~sup>(Ω) are equivalent:~~

	~~:<math>\\| \cdot \\| : H^{k} (\Omega) \to \mathbf{R}: u \mapsto \\| u \\| := \sqrt{\sum_{\| \alpha \| \leq k} \\| \mathrm{D}^{\alpha} u \\|_{L^{2} (\Omega)}^{2}}<~~/~~math>~~

	~~and~~

	~~:<math>\\| \cdot \\|' : H^{k} (\Omega) \to \mathbf{R}: u \mapsto \\| u \\|' := \sqrt{\\| u \\|_{L^{1} (\Omega)}^{2} + \sum_{\| \alpha \| = k} \\| \mathrm{D}^{\alpha} u \\|_{L^{2} (\Omega)}^{2}}.</math>~~

	For the subspace of ''H''<sup>''k''</sup>(Ω) consisting of those Sobolev functions with [[trace operator\|zero trace]] (those that are "zero on the boundary" of Ω), the ''L''<sup>1</sup> norm of ''u'' can be left out to yield another equivalent norm.

	~~==References==~~

	* {{cite book
	~~\| last1 = Renardy~~
	~~\| first1 = Michael~~
	~~\| last2 = Rogers~~
	~~\| first2 = Robert C.~~
	~~\| title = An Introduction to Partial Differential Equations~~
	~~\| publisher = Springer~~-~~Verlag~~
	~~\| location = Berlin~~
	~~\| year=1992~~
	~~\| isbn=978~~-3-~~540~~-~~97952~~-4
	}}

	~~[[Category:Banach spaces~~]]
	~~[[Category:Sobolev spaces]]~~
	~~[[Category:Lemmas]]~~

	~~{{mathanalysis-stub}}~~