Multiplicity-one theorem - Revision history

en>BG19bot: /* Definition */WP:CHECKWIKI error fix for #98. Broken sub tag. Do general fixes if a problem exists. -, replaced: _{π → _π using AWB (9957)}

2014-03-01T09:21:29Z

Definition: WP:CHECKWIKI error fix for #98. Broken sub tag. Do general fixes if a problem exists. -, replaced: π → π using AWB (9957)

← Older revision		Revision as of 11:21, 1 March 2014
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	~~In [[real analysis]], a branch of mathematics, '''Cousin's theorem''' states that:~~		Hi also there. Let me begin with introducing the author, her name is Angele. Since I was 18 I've been working to be a financial expert. One of stuff she loves most is reading comics but she can't insure that it is her group. She's always loved living in florida and she loves a day living high. You can find my website here: http://www.medicalgroup-cerro.it/scarpe-hogan/<br><br>my web site: [http://www.medicalgroup-cerro.it/scarpe-hogan/ hogan scarpe]

	~~:If for every point of a closed region (in modern terms, "[[closed set\|closed]] and [[bounded (set theory)\|bounded]]")~~ there ~~is a circle of finite radius (in modern term~~, ~~a "[[neighbourhood (mathematics)\|neighborhood]]") , then the region can be divided into a finite number of subregions such that each subregion~~ is ~~interior~~ to a ~~circle of a given set having its center in the subregion~~.~~<ref name="h1">Hildebrandt 1925, p. 29</ref>~~

	~~This result was proved and established by Pierre Cousin, a student~~ of ~~[[Henri Poincaré]], in 1895, and~~ it is an extension of the original [[Heine–Borel theorem]] on [[compactness]] for arbitrary [[set cover\|covers]] of any [[compact space\|compact]] subsets of <math>\mathbb{R}^n</math>. However, Pierre Cousin did not receive any credit. ~~'''Cousin~~'s ~~theorem''' was generally attributed to [[Henri Lebesgue]] and renamed as '''Borel–Lebesgue theorem''', who was aware of this result~~ in ~~1898~~ and ~~proved this in his dissertation in 1903~~.~~<ref name="h1"/>~~

	~~Nowadays, it is stated as~~:
	:Let <math>\mathcal{C}</math> be a full cover of [''a'', ''b''], that is, a collection of closed subintervals of [''a'', ''b''] with the property that for every ''x''∈[''a'', ''b''], there exists a ''δ''>0 so that <math>\mathcal{C}</~~math> contains all subintervals of [''a'', ''b''] which contains ''x'' and length smaller than ''δ''. Then there exists a partition {''I<sub>1</sub>'', ''I<sub>2<~~/~~sub>'',~~..~~.,''I<sub>n</sub>''} of non-overlapping intervals for [''a'', ''b''], where ''I<sub>i<~~/~~sub>''=[''x<sub>i~~-1</~~sub>'', ''x~~<~~sub~~>i<~~/sub~~>~~'']∈<math>\mathcal{C}</math> and ''a=x<sub>0</sub> < x<sub>1</sub> <...< x<sub>n</sub>=b'' for all ''1≤i≤n''.~~

	~~Further, '''Cousin's theorem''' is mainly only used in [[Henstock–Kurzweil integral]] and is often called '''Fineness Theorem''' or [~~[~~Cousin's lemma]]. It can be stated as:~~
	:~~If ''I'' := [''a'', ''b''] ⊆ '''R'''<sup>''n''<~~/sup> is a [[degeneracy (mathematics)\|nondegenerate]] [[compact space\|compact]] interval and ''δ'' is any gauge defined on ''I'', then there always exists a tagged partition of ''I'' that is ''δ''-fine.<ref name="b1">Bartle 2001, p. 11</~~ref>~~

	~~==Notes==~~
	~~{{Reflist}}~~

	~~==References==~~

	~~{{refbegin}}~~
	*Hildebrandt, T. H. (1925). ''The Borel Theorem and its Generalizations'' In J. C. Abbott (Ed.~~), The Chauvenet Papers: A collection of Prize~~-~~Winning Expository Papers in Mathematics~~. ~~Mathematical Association of America.~~
	*Raman, M. J. (1997). ''Understanding Compactness: A Historical Perspective'', Master of Arts Thesis. University of California, Berkeley.
	*Bartle, R. G. (2001). ''A Modern Theory of Integration'', Graduate Studies in Mathematics '''32''', American Mathematical Society.
	~~{{refend}}~~


	~~{{mathanalysis~~-~~stub}}~~



	~~[[Category:Real analysis]~~]

en>MZMcBride: -:Category:Article Feedback 5

2012-12-24T06:59:01Z

-Category:Article Feedback 5

← Older revision		Revision as of 08:59, 24 December 2012
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	~~With the backdrop of the Superstition Mountains~~, ~~you have~~ a ~~scenic mountain bike trail network that can embody every quality~~ of off-road riding in the Southwest. When you have just about any queries regarding in which along with the best way to employ [http://originsofttech.net/AgaArcade/profile/librain Cannondale mountain bike sizing.], ~~you can call us in our own internet site. You~~'ll ask yourself this question in conjunction with "how much will you pay. High end mountain bikes use carbon fibre frames, or other, more exotic materials to reduce weight and keep stiffness up. Generally speaking, a good, durable mountain bike starts at about $600 and goes up from there. Look at what other riders appreciate about their bikes. <br><br>		In [[real analysis]], a branch of mathematics, '''Cousin's theorem''' states that:

	~~The prices offered by these online stores are very competitive~~ and ~~realistic according to the market.~~ in ~~summary let me give some procuations you bshould take when ridding it~~ , ~~ridding~~ a ~~bike at high speed~~ can ~~lead~~ to ~~accidents just maintain~~ a ~~minimum~~ and ~~also follow the traffic lights~~ , if it is ~~for sport just make enough practice and enjoy~~ the ~~game. The price you can expect to pay~~ for ~~stems is actually determined by the materials used along with its weight. More than just the excitement of biking you can be delighted by the beautiful whistle~~ of ~~a hill myna or by the amicable smile~~ of ~~a villager~~. ~~Then sense the direct route it will take~~ to ~~go from your nose to the target location~~ and ~~then back out through the nose~~. <br><br>~~The "derailleur" attached to the gear shifter moves the chain from one ring to another. An Indian travois style trailer~~ is ~~very simple to make and very effective for heavy loads. The blogs and news feeds can also be shared among the friends and family free~~ of ~~cost via the social media networks. Once you have answered the above questions and you see all these features present in~~ a ~~bike~~, ~~you are on your way to choosing~~ the ~~road bike~~ that ~~is right~~ for ~~you~~. ~~The most common sized crankset is the 170 millimeter~~. <br><br>~~With knowledge on the strengths and weakness~~ of ~~the MTB model~~, ~~they are~~ a ~~great source of help~~. ~~The more bikes you can test drive~~, ~~you better you~~'~~ll understand what works~~ and ~~what doesn~~'t. It is a ~~quality piece available at affordable price. People like to ride the mountain bike to those terrains because they enjoy the thrill~~ and ~~the adventure. To find out the latest News and Tips~~ on ~~folding bikes~~, ~~visit Best Folding Bikes at bestfoldingbikes~~. <br><br>A ~~small spot~~ of ~~rust can quickly grow and cause weak areas on your frame~~. ~~If you are cycling on incredibly tough uneven ground through woods~~, ~~up mountains etc you would need a full suspension mountain bike~~. ~~This may be related to a muscle imbalance between opposing muscle groups in the leg and is commonly related to excessive foot pronation~~ (~~collapsing arch~~). ~~Mountain bike wheels provide a good mixture~~ of ~~traction~~, ~~stability~~, ~~and sturdiness ; and different riding conditions need specific types~~ of ~~mountain bicycle wheels. If they connect with the wrong part~~, ~~it could cause you to stop on a dime~~, ~~which will most likely result in you taking a tumble over the handlebars~~.		:If for every point of a closed region (in modern terms, "[[closed set\|closed]] and [[bounded (set theory)\|bounded]]") there is a circle of finite radius (in modern term, a "[[neighbourhood (mathematics)\|neighborhood]]") , then the region can be divided into a finite number of subregions such that each subregion is interior to a circle of a given set having its center in the subregion.<ref name="h1">Hildebrandt 1925, p. 29</ref>

			This result was proved and established by Pierre Cousin, a student of [[Henri Poincaré]], in 1895, and it is an extension of the original [[Heine–Borel theorem]] on [[compactness]] for arbitrary [[set cover\|covers]] of any [[compact space\|compact]] subsets of <math>\mathbb{R}^n</math>. However, Pierre Cousin did not receive any credit. '''Cousin's theorem''' was generally attributed to [[Henri Lebesgue]] and renamed as '''Borel–Lebesgue theorem''', who was aware of this result in 1898 and proved this in his dissertation in 1903.<ref name="h1"/>

			Nowadays, it is stated as:
			:Let <math>\mathcal{C}</math> be a full cover of [''a'', ''b''], that is, a collection of closed subintervals of [''a'', ''b''] with the property that for every ''x''∈[''a'', ''b''], there exists a ''δ''>0 so that <math>\mathcal{C}</math> contains all subintervals of [''a'', ''b''] which contains ''x'' and length smaller than ''δ''. Then there exists a partition {''I<sub>1</sub>'', ''I<sub>2</sub>'',...,''I<sub>n</sub>''} of non-overlapping intervals for [''a'', ''b''], where ''I<sub>i</sub>''=[''x<sub>i-1</sub>'', ''x<sub>i</sub>'']∈<math>\mathcal{C}</math> and ''a=x<sub>0</sub> < x<sub>1</sub> <...< x<sub>n</sub>=b'' for all ''1≤i≤n''.

			Further, '''Cousin's theorem''' is mainly only used in [[Henstock–Kurzweil integral]] and is often called '''Fineness Theorem''' or [[Cousin's lemma]]. It can be stated as:
			:If ''I'' := [''a'', ''b''] ⊆ '''R'''<sup>''n''</sup> is a [[degeneracy (mathematics)\|nondegenerate]] [[compact space\|compact]] interval and ''δ'' is any gauge defined on ''I'', then there always exists a tagged partition of ''I'' that is ''δ''-fine.<ref name="b1">Bartle 2001, p. 11</ref>

			==Notes==
			{{Reflist}}

			==References==

			{{refbegin}}
			*Hildebrandt, T. H. (1925). ''The Borel Theorem and its Generalizations'' In J. C. Abbott (Ed.), The Chauvenet Papers: A collection of Prize-Winning Expository Papers in Mathematics. Mathematical Association of America.
			*Raman, M. J. (1997). ''Understanding Compactness: A Historical Perspective'', Master of Arts Thesis. University of California, Berkeley.
			*Bartle, R. G. (2001). ''A Modern Theory of Integration'', Graduate Studies in Mathematics '''32''', American Mathematical Society.
			{{refend}}


			{{mathanalysis-stub}}



			[[Category:Real analysis]]

en>GoingBatty: /* Definition */bad link repair using AWB (8023)

2012-03-18T21:28:55Z

Definition: bad link repair using AWB (8023)

New page

With the backdrop of the Superstition Mountains, you have a scenic mountain bike trail network that can embody every quality of off-road riding in the Southwest. When you have just about any queries regarding in which along with the best way to employ [http://originsofttech.net/AgaArcade/profile/librain Cannondale mountain bike sizing.], you can call us in our own internet site. You'll ask yourself this question in conjunction with "how much will you pay. High end mountain bikes use carbon fibre frames, or other, more exotic materials to reduce weight and keep stiffness up. Generally speaking, a good, durable mountain bike starts at about $600 and goes up from there. Look at what other riders appreciate about their bikes. 

The prices offered by these online stores are very competitive and realistic according to the market. in summary let me give some procuations you bshould take when ridding it , ridding a bike at high speed can lead to accidents just maintain a minimum and also follow the traffic lights , if it is for sport just make enough practice and enjoy the game. The price you can expect to pay for stems is actually determined by the materials used along with its weight. More than just the excitement of biking you can be delighted by the beautiful whistle of a hill myna or by the amicable smile of a villager. Then sense the direct route it will take to go from your nose to the target location and then back out through the nose. The "derailleur" attached to the gear shifter moves the chain from one ring to another. An Indian travois style trailer is very simple to make and very effective for heavy loads. The blogs and news feeds can also be shared among the friends and family free of cost via the social media networks. Once you have answered the above questions and you see all these features present in a bike, you are on your way to choosing the road bike that is right for you. The most common sized crankset is the 170 millimeter. With knowledge on the strengths and weakness of the MTB model, they are a great source of help. The more bikes you can test drive, you better you'll understand what works and what doesn't. It is a quality piece available at affordable price. People like to ride the mountain bike to those terrains because they enjoy the thrill and the adventure. To find out the latest News and Tips on folding bikes, visit Best Folding Bikes at bestfoldingbikes. A small spot of rust can quickly grow and cause weak areas on your frame. If you are cycling on incredibly tough uneven ground through woods, up mountains etc you would need a full suspension mountain bike. This may be related to a muscle imbalance between opposing muscle groups in the leg and is commonly related to excessive foot pronation (collapsing arch). Mountain bike wheels provide a good mixture of traction, stability, and sturdiness ; and different riding conditions need specific types of mountain bicycle wheels. If they connect with the wrong part, it could cause you to stop on a dime, which will most likely result in you taking a tumble over the handlebars.

Multiplicity-one theorem - Revision history

en>BG19bot: /* Definition */WP:CHECKWIKI error fix for #98. Broken sub tag. Do general fixes if a problem exists. -, replaced: π → π using AWB (9957)

en>MZMcBride: -:Category:Article Feedback 5

en>GoingBatty: /* Definition */bad link repair using AWB (8023)

en>BG19bot: /* Definition */WP:CHECKWIKI error fix for #98. Broken sub tag. Do general fixes if a problem exists. -, replaced: _{π → _π using AWB (9957)}