Permanent - Revision history

en>Wcherowi: Reverted 1 edit by Medical Historian III (talk) to last revision by Rjwilmsi. (TW)

2014-08-21T16:15:22Z

Reverted 1 edit by Medical Historian III (talk) to last revision by Rjwilmsi. (TW)

← Older revision		Revision as of 18:15, 21 August 2014
Line 1:		Line 1:
	In [[mathematics]], the '''Hilbert cube''', named after [[David Hilbert]], is a [[topological space]] that provides an instructive example of some ideas in [[topology]]. Furthermore, many interesting topological spaces can be embedded in the Hilbert cube; that is, can be viewed as subspaces of the Hilbert cube (see below).		Nothing to tell about me really.<br>Nice to be here and a part of this site.<br>I really wish I am useful at all<br><br>Here is my site; [http://hemorrhoidtreatmentfix.com/bleeding-hemorrhoids-treatment bleeding hemorrhoids treatment]

	~~==Definition==~~
	The Hilbert cube is best defined as the [[topological product]] of the [[interval (mathematics)\|intervals]] [0, 1/''n''] for ''n'' = 1, 2, 3, 4, ... That is, it is a [[cuboid]] of [[countably infinite]] [[dimension]], where the lengths of the edges in each orthogonal direction form the sequence <math>\lbrace 1/n \rbrace_{n\in\mathbb{N}}</math>.

	~~The Hilbert cube is [[homeomorphism\|homeomorphic]]~~ to ~~the product of [[countably infinite]]ly many copies of the [[unit interval]] [0, 1]~~. ~~In other words, it is topologically indistinguishable from the [[unit cube]] of countably infinite dimension.~~

	~~If a point in the Hilbert cube is specified by a sequence <math>\lbrace a_n \rbrace</math> with~~ <~~math>0 \leq a_n \leq 1/n</math~~>~~, then a homeomorphism~~ to ~~the infinite dimensional unit cube is given by <math>h : a_n \rarr n\cdot a_n</math>.~~

	~~==The Hilbert cube as~~ a ~~metric space==~~
	~~It is sometimes convenient to think~~ of ~~the Hilbert cube as a [[metric space]], indeed as a specific subset of a separable [[Hilbert space]] (i.e. a Hilbert space with a countably infinite Hilbert basis).~~
	~~For these purposes, it is best not to think of it as a product of copies of [0,1], but instead as~~

	~~:[0,1] × [0,1/2] × [0,1/3] × ···;~~

	~~as stated above, for topological properties,~~ this ~~makes no difference.~~
	~~That is, an element of the Hilbert cube is an [[infinite sequence]]~~

	~~:(''x''<sub>''n''</sub>)~~

	~~that satisfies~~

	~~:0 ≤ ''x''<sub>''n''</sub> ≤ 1/''n''.~~

	Any such sequence belongs to the Hilbert space [[Lp_space#The_p-norm_in_countably_infinite_dimensions\|ℓ<sub>2</sub>]], so the Hilbert cube inherits a metric from there. One can show that the topology induced by the metric is the same as the [[product topology]] in the above definition.

	~~==Properties==~~
	~~As a product of [[compact (topology)\|compact]] [[Hausdorff space]]s, the Hilbert cube is itself a compact Hausdorff space as a result of the [[Tychonoff theorem]]~~.
	~~The compactness of the Hilbert cube can also be proved without the Axiom of Choice by constructing a continuous function from the usual [[Cantor set]] onto the Hilbert cube.~~

	~~In ℓ~~<~~sub~~>~~2</sub>, no point has a compact [[neighbourhood (topology)\|neighbourhood]] (thus, ℓ<sub>2</sub> is not [[locally compact]]). One might expect that~~ all ~~of the compact subsets of ℓ~~<~~sub~~>2<~~/sub~~> ~~are finite-dimensional.~~
	~~The Hilbert cube shows that this~~ is ~~not the case.~~
	But the Hilbert cube fails to be a neighbourhood of any point ''p'' because its side becomes smaller and smaller in each dimension, so that an [[open ball]] around ''p'' of any fixed radius ''e'' > 0 must go outside the cube in some dimension.

	Every subset of the Hilbert cube inherits from the Hilbert cube the properties of being both metrizable (and therefore [[Normal space\|T4]]) and [[second countable]]. It is more interesting that the converse also holds: Every [[second countable]] [[Normal space\|T4]] space is homeomorphic to a subset of the Hilbert cube.

	~~Every G<sub>δ</sub>-subset of the Hilbert cube is a [[Polish space]], a topological space homeomorphic to a separable and complete metric space. Conversely, every Polish space is homeomorphic to a~~ [~~[Gδ set\|G<sub>δ</sub>-subset]] of the Hilbert cube.<ref>[[#Srivastava\|Srivastava]], pp. 55</ref>~~

	~~== Notes ==~~
	~~{{reflist\|2}}~~

	~~== References ==~~
	* <cite id="Srivastava">{{Cite book \| last = Srivastava \| first = Sashi Mohan \| authorlink = \| title = A Course on Borel Sets \| url = http://~~books.google~~.com/~~?id=FhYGYJtMwcUC \| accessdate = 12-04-08 \| publisher = [[Springer-Verlag]] \| series = Graduate Texts in Mathematics \| year = 1998 \| doi = \| isbn = 978-0-387-98412-4 }}</cite>~~
	* {{Cite book \| last1=Steen \| first1=Lynn Arthur \| author1-link=Lynn Arthur Steen \| last2=Seebach \| first2=J. Arthur Jr. \| author2-link=J. Arthur Seebach, Jr. \| title=[[Counterexamples in Topology]] \| origyear=1978 \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| edition=[[Dover Publications\|Dover]] reprint of 1978 \| isbn=978-0-~~486-68735-3 \| id={{MathSciNet\|id=507446}} \| year=1995 \| postscript=<!--None-->}}~~
	~~[[Category:Topological spaces]~~]

en>Wcherowi: /* Perfect matchings */ wikilink

2013-12-12T16:39:28Z

Perfect matchings: wikilink

← Older revision		Revision as of 18:39, 12 December 2013
Line 1:		Line 1:
	~~Taking~~ some ~~diet plans to lose fat~~ is ~~not convenient~~. ~~Some are successful but several are failed. While along~~ the ~~way~~ of ~~dieting~~, ~~you need to sacrifice~~ for ~~the unique we so we could feel deprived~~. ~~However~~, ~~permanent fat loss plus healthy body without suffering will be attained that~~ is ~~something I would like to share with you here.~~<br><br>~~So, again, back~~ to the ~~all-natural aspect;~~ many ~~persons have found~~ [~~https://safedietsthatwork~~.~~shutterfly~~.~~com~~/~~22 diet plans] effectively is replacing the regular food~~ with ~~natural replacements. These are clearly marked; usually not containing pesticides~~, ~~artificial flavoring, or preservatives. Doctors might tell you which organic is the method~~ to ~~go, however, some people create~~ the ~~mistake of thinking they could just gorge themselves on organic food plus everything~~ is ~~fine. That's no method to follow, even diets which work.~~<br><br>~~Similar~~ to the ~~1200 calorie diet~~, a ~~1500 calorie diet is a diet which limits the amount~~ of ~~calories to be consumed inside~~ a ~~day, to 1500~~. ~~Two such sample diet plans are provided below inside~~ a ~~tabular form~~.~~<br><br>Then~~, ~~there~~ is ~~the meal delivery or prepackaged diet plan. This enables you~~ to ~~purchase prepackaged food. As an example in the event you purchase your~~ 3 ~~meals and snacks~~ for ~~a day~~. ~~This~~ is ~~very challenging~~, ~~will be powerful for several time period nevertheless could not be perfect for a long expression goal considering you might not wish To eat~~ the ~~prepackaged foods all your existence~~.<br><br>~~All diets are built about~~ the ~~concept that~~ there ~~are foods a dieter cannot have~~. ~~Deprivation leads to feelings of hunger and~~ that ~~creates powerful cravings for~~ the ~~foods that are taboo It~~ is the same ~~concept because telling~~ a ~~child not to touch anything considering it happens to~~ be ~~hot~~.<br><br>~~Most criticism of low carbohydrate dieting~~ is ~~based on the fact~~ that ~~the dieters will get nearly~~ all of ~~their daily calories from meat, dairy plus other excellent fat products. Medical advice is advised before starting this sort~~ of ~~diet.~~<br><br>~~High consumption of carbohydrates~~ is the ~~primary reason for rising weight statistics across the planet~~. ~~Many of~~ the ~~junk foods are excessively high inside carbohydrates, that have~~ to be ~~immediately eliminated when one desires to have~~ a ~~healthy lifetime. Although you are planning to start with a diet~~, ~~it really is necessary~~ that ~~you continue with right consumption~~ of ~~important nutrients~~. ~~Do not rush into a no carb diet, before consulting~~ the ~~dietitian~~. ~~Starving~~ is the ~~largest mistake folks create whilst trying~~ to ~~lose fat~~. ~~Changing food habits is an necessary piece~~ of ~~staying fit. However, along with changing~~ the ~~diet~~, ~~one has~~ to ~~exercise~~ to ~~get into shape. After~~ a set ~~exercise routine and preserving a diet regime might assist a person reside a healthier lifetime~~.		In [[mathematics]], the '''Hilbert cube''', named after [[David Hilbert]], is a [[topological space]] that provides an instructive example of some ideas in [[topology]]. Furthermore, many interesting topological spaces can be embedded in the Hilbert cube; that is, can be viewed as subspaces of the Hilbert cube (see below).

			==Definition==
			The Hilbert cube is best defined as the [[topological product]] of the [[interval (mathematics)\|intervals]] [0, 1/''n''] for ''n'' = 1, 2, 3, 4, ... That is, it is a [[cuboid]] of [[countably infinite]] [[dimension]], where the lengths of the edges in each orthogonal direction form the sequence <math>\lbrace 1/n \rbrace_{n\in\mathbb{N}}</math>.

			The Hilbert cube is [[homeomorphism\|homeomorphic]] to the product of [[countably infinite]]ly many copies of the [[unit interval]] [0, 1]. In other words, it is topologically indistinguishable from the [[unit cube]] of countably infinite dimension.

			If a point in the Hilbert cube is specified by a sequence <math>\lbrace a_n \rbrace</math> with <math>0 \leq a_n \leq 1/n</math>, then a homeomorphism to the infinite dimensional unit cube is given by <math>h : a_n \rarr n\cdot a_n</math>.

			==The Hilbert cube as a metric space==
			It is sometimes convenient to think of the Hilbert cube as a [[metric space]], indeed as a specific subset of a separable [[Hilbert space]] (i.e. a Hilbert space with a countably infinite Hilbert basis).
			For these purposes, it is best not to think of it as a product of copies of [0,1], but instead as

			:[0,1] × [0,1/2] × [0,1/3] × ···;

			as stated above, for topological properties, this makes no difference.
			That is, an element of the Hilbert cube is an [[infinite sequence]]

			:(''x''<sub>''n''</sub>)

			that satisfies

			:0 ≤ ''x''<sub>''n''</sub> ≤ 1/''n''.

			Any such sequence belongs to the Hilbert space [[Lp_space#The_p-norm_in_countably_infinite_dimensions\|ℓ<sub>2</sub>]], so the Hilbert cube inherits a metric from there. One can show that the topology induced by the metric is the same as the [[product topology]] in the above definition.

			==Properties==
			As a product of [[compact (topology)\|compact]] [[Hausdorff space]]s, the Hilbert cube is itself a compact Hausdorff space as a result of the [[Tychonoff theorem]].
			The compactness of the Hilbert cube can also be proved without the Axiom of Choice by constructing a continuous function from the usual [[Cantor set]] onto the Hilbert cube.

			In ℓ<sub>2</sub>, no point has a compact [[neighbourhood (topology)\|neighbourhood]] (thus, ℓ<sub>2</sub> is not [[locally compact]]). One might expect that all of the compact subsets of ℓ<sub>2</sub> are finite-dimensional.
			The Hilbert cube shows that this is not the case.
			But the Hilbert cube fails to be a neighbourhood of any point ''p'' because its side becomes smaller and smaller in each dimension, so that an [[open ball]] around ''p'' of any fixed radius ''e'' > 0 must go outside the cube in some dimension.

			Every subset of the Hilbert cube inherits from the Hilbert cube the properties of being both metrizable (and therefore [[Normal space\|T4]]) and [[second countable]]. It is more interesting that the converse also holds: Every [[second countable]] [[Normal space\|T4]] space is homeomorphic to a subset of the Hilbert cube.

			Every G<sub>δ</sub>-subset of the Hilbert cube is a [[Polish space]], a topological space homeomorphic to a separable and complete metric space. Conversely, every Polish space is homeomorphic to a [[Gδ set\|G<sub>δ</sub>-subset]] of the Hilbert cube.<ref>[[#Srivastava\|Srivastava]], pp. 55</ref>

			== Notes ==
			{{reflist\|2}}

			== References ==
			* <cite id="Srivastava">{{Cite book \| last = Srivastava \| first = Sashi Mohan \| authorlink = \| title = A Course on Borel Sets \| url = http://books.google.com/?id=FhYGYJtMwcUC \| accessdate = 12-04-08 \| publisher = [[Springer-Verlag]] \| series = Graduate Texts in Mathematics \| year = 1998 \| doi = \| isbn = 978-0-387-98412-4 }}</cite>
			* {{Cite book \| last1=Steen \| first1=Lynn Arthur \| author1-link=Lynn Arthur Steen \| last2=Seebach \| first2=J. Arthur Jr. \| author2-link=J. Arthur Seebach, Jr. \| title=[[Counterexamples in Topology]] \| origyear=1978 \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| edition=[[Dover Publications\|Dover]] reprint of 1978 \| isbn=978-0-486-68735-3 \| id={{MathSciNet\|id=507446}} \| year=1995 \| postscript=<!--None-->}}
			[[Category:Topological spaces]]

en>Jim1138: Reverted edits by 120.28.98.107 (talk) to last revision by 79.222.3.215 (HG)

2012-08-22T05:17:58Z

Reverted edits by 120.28.98.107 (talk) to last revision by 79.222.3.215 (HG)

New page

Taking some diet plans to lose fat is not convenient. Some are successful but several are failed. While along the way of dieting, you need to sacrifice for the unique we so we could feel deprived. However, permanent fat loss plus healthy body without suffering will be attained that is something I would like to share with you here. So, again, back to the all-natural aspect; many persons have found [https://safedietsthatwork.shutterfly.com/22 diet plans] effectively is replacing the regular food with natural replacements. These are clearly marked; usually not containing pesticides, artificial flavoring, or preservatives. Doctors might tell you which organic is the method to go, however, some people create the mistake of thinking they could just gorge themselves on organic food plus everything is fine. That's no method to follow, even diets which work. Similar to the 1200 calorie diet, a 1500 calorie diet is a diet which limits the amount of calories to be consumed inside a day, to 1500. Two such sample diet plans are provided below inside a tabular form. Then, there is the meal delivery or prepackaged diet plan. This enables you to purchase prepackaged food. As an example in the event you purchase your 3 meals and snacks for a day. This is very challenging, will be powerful for several time period nevertheless could not be perfect for a long expression goal considering you might not wish To eat the prepackaged foods all your existence. All diets are built about the concept that there are foods a dieter cannot have. Deprivation leads to feelings of hunger and that creates powerful cravings for the foods that are taboo It is the same concept because telling a child not to touch anything considering it happens to be hot. Most criticism of low carbohydrate dieting is based on the fact that the dieters will get nearly all of their daily calories from meat, dairy plus other excellent fat products. Medical advice is advised before starting this sort of diet. High consumption of carbohydrates is the primary reason for rising weight statistics across the planet. Many of the junk foods are excessively high inside carbohydrates, that have to be immediately eliminated when one desires to have a healthy lifetime. Although you are planning to start with a diet, it really is necessary that you continue with right consumption of important nutrients. Do not rush into a no carb diet, before consulting the dietitian. Starving is the largest mistake folks create whilst trying to lose fat. Changing food habits is an necessary piece of staying fit. However, along with changing the diet, one has to exercise to get into shape. After a set exercise routine and preserving a diet regime might assist a person reside a healthier lifetime.