Grassmannian - Revision history

en>John of Reading: Typo/general fixing, replaced: vice-versa → vice versa using AWB

2015-01-04T14:29:57Z

Typo/general fixing, replaced: vice-versa → vice versa using AWB

99.241.166.168: /* The Grassmannian as a scheme */ improved formatting and wording

2014-02-19T00:30:46Z

The Grassmannian as a scheme: improved formatting and wording

← Older revision		Revision as of 02:30, 19 February 2014
Line 1:		Line 1:
	~~In [[mathematics]], '''limit cardinals''' are certain [[cardinal number]]s. A cardinal number λ~~ is ~~a '''weak limit cardinal''' if λ is neither a [[successor cardinal]] nor zero. This means~~ that ~~one cannot "reach" λ by repeated successor operations~~. ~~These cardinals are sometimes called simply "limit cardinals" when the context~~ is ~~clear.~~		Wilber Berryhill is the name his parents gave him and he completely digs that title. My working day occupation is an info officer but I've already applied for another one. The preferred pastime for him and his children is style and he'll be starting some thing else alongside with it. Alaska is exactly where I've usually been residing.<br><br>Review my blog: psychic phone readings - [http://www.clubwww1music.com/profile.php?u=ImOFO www.clubwww1music.com this guy],

	~~A cardinal λ is a~~ '~~''strong limit cardinal''' if λ cannot be reached by repeated [[Power set\|powerset]] operations. This means that λ is nonzero and,~~ for ~~all κ < λ, 2<sup>κ</sup> < λ~~. ~~Every strong limit cardinal is also a weak limit cardinal, because κ<sup>+</sup> ≤ 2<sup>κ</sup>~~ for ~~every cardinal κ, where κ<sup>+</sup> denotes the successor cardinal of κ.~~

	~~The first infinite cardinal, <math>\aleph_0</math> ([[Aleph number#Aleph-naught\|aleph-naught]]), is a strong limit cardinal,~~ and ~~hence also a weak limit cardinal.~~

	~~== Constructions ==~~

	~~One way to construct limit cardinals~~ is ~~via the union operation: <math>\aleph_{\omega}</math> is a weak limit cardinal, defined as the union of all the alephs before it;~~ and ~~in general <math>\aleph_{\lambda}</math> for any [[limit ordinal]] λ is a weak limit cardinal.~~

	~~The [[beth number\|ב operation]] can~~ be ~~used to obtain strong limit cardinals. This operation is a map from ordinals to cardinals defined as~~
	~~:<math>\beth_{0} = \aleph_0,</math>~~
	~~:<math>\beth_{\alpha+1} = 2^{\beth_{\alpha}},</math> (the smallest ordinal [[equinumerous]]~~ with ~~the powerset)~~
	~~:If λ is a limit ordinal, <math>\beth_{\lambda} = \bigcup \{ \beth_{\alpha} : \alpha < \lambda\}.</math>~~
	~~The cardinal~~
	~~:<math>\beth_{\omega} = \bigcup \{ \beth_{0}, \beth_{1}, \beth_{2}, \ldots \} = \bigcup_{n < \omega} \beth_{n} </math>~~
	~~is a strong limit cardinal of [[cofinality]] ω. More generally, given any ordinal α, the cardinal~~
	~~:<math>\beth_{\alpha+\omega} = \bigcup_{n < \omega} \beth_{\alpha+n} </math>~~
	~~is a strong limit cardinal. Thus there are arbitrarily large strong limit cardinals.~~

	~~== Relationship with ordinal subscripts ==~~

	If the [[axiom of choice]] holds, every cardinal number has an [[initial ordinal]]. If that initial ordinal is <math>\omega_{\lambda} \,,</math> then the cardinal number is of the form <math>\aleph_\lambda</math> for the same ordinal subscript λ. The ordinal λ determines whether <math>\aleph_\lambda</math> is a weak limit cardinal. Because <math>\aleph_{\alpha^+} = (\aleph_\alpha)^+ \,,</math> if λ is a successor ordinal then <math>\aleph_\lambda</math> is not a weak limit. Conversely, if a cardinal κ is a successor cardinal, say <math>\kappa = (\aleph_{\alpha})^+ \,,</math> then <math>\kappa = \aleph_{\alpha^+} \,.</math> Thus, in general, <math>\aleph_\lambda</math> is a weak limit cardinal if and only if λ is zero or a limit ordinal.

	~~Although the ordinal subscript tells whether a cardinal is a weak limit,~~ it ~~does not tell whether a cardinal is a strong limit~~. ~~For example, [[Zermelo–Fraenkel set theory\|ZFC]] proves that <math>\aleph_\omega</math>~~ is ~~a weak limit cardinal, but neither proves nor disproves that <math>\aleph_\omega</math> is a strong limit cardinal (Hrbacek and Jech 1999:168)~~. ~~The [[generalized continuum hypothesis]] states that~~ <~~math~~>~~\kappa^+ = 2^{\kappa} \,~~<~~/math~~> ~~for every infinite cardinal κ. Under this hypothesis, the notions of weak and strong limit cardinals coincide.~~

	~~== The notion of inaccessibility and large cardinals ==~~

	~~The preceding defines a notion of "inaccessibility"~~: we are dealing with cases where it is no longer enough to do finitely many iterations of the successor and powerset operations; hence the phrase "cannot be reached" in both of the intuitive definitions above. But the "union operation" always provides another way of "accessing" these cardinals (and indeed, such is the case of limit ordinals as well). Stronger notions of inaccessibility can be defined using [[cofinality]]. For a weak (resp. strong) limit cardinal κ the requirement that cf(κ) = κ (i.e. κ be [[regular cardinal\|regular]]) so that κ cannot be expressed as a sum (union) of fewer than κ smaller cardinals. Such a cardinal is called a [[inaccessible cardinal\|weakly (resp. strongly) inaccessible cardinal]]. The preceding examples both are singular cardinals of cofinality ω and hence they are not inaccessible.

	~~<math>\aleph_0</math> would be an inaccessible cardinal of both "strengths" except that the definition of inaccessible requires that they be uncountable. Standard Zermelo~~-~~Fraenkel set theory with the Axiom of Choice (ZFC) cannot even prove the consistency of the existence of an inaccessible cardinal of either kind above <math>\aleph_0</math>, due to~~ [~~[Gödel's incompleteness theorems\|Gödel's Incompleteness Theorem]]. More specifically, if <math>\kappa<~~/~~math> is weakly inaccessible then <math>L_{\kappa} \models ZFC<~~/~~math>~~. ~~These form the first in a hierarchy of [[large cardinal]]s~~.

	~~== See also ==~~

	* [[Cardinal number]]

	~~== References ==~~
	* {{citation
	~~\| last1= Hrbacek \| first1 = Karel~~
	~~\| last2=Jech \| first2=Thomas~~
	~~\| title = Introduction to Set Theory~~
	~~\| edition = 3~~
	~~\| year = 1999 \| isbn = 0-8247-7915-0~~
	}}
	* {{Citation \| last1=Jech \| first1=Thomas \| author1-link=Thomas Jech \| title=Set Theory \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| edition=third millennium \| series=Springer Monographs in Mathematics \| isbn=978-3-540-44085-7 \| doi=10.~~1007/3-540-44761-X \| year=2003}}~~
	* {{Citation \| last1=Kunen \| first1=Kenneth \| author1-link=Kenneth Kunen \| title=Set theory: An introduction to independence proofs \| publisher=~~[[Elsevier]] \| isbn=978-0-444-86839-8 \| year=1980}}~~

	~~==External links==~~

	* http://www.ii.com~~/math/cardinals/ Infinite ink on cardinals~~

	~~[[Category:Set theory]]~~
	~~[[Category:Cardinal numbers]~~]

en>Joel B. Lewis: /* The Grassmannian as a set */

2014-01-07T15:04:48Z

The Grassmannian as a set

← Older revision		Revision as of 17:04, 7 January 2014
Line 1:		Line 1:
	~~The action of pedaling~~ a ~~bicycle aids in weight reduction~~. ~~Uphill cycling plus high strength cycling is an great~~ means ~~of developing~~ the ~~muscles in the body to create them stronger~~. ~~At the same time it burns fat too~~. ~~You need both to happen~~ and ~~cycling accomplishes this~~.<br><br>~~On the other hand~~, ~~overweight people who usually [http:~~/~~/safedietplans~~.~~com~~/~~bmr~~-~~calculator bmr calculator~~] ~~try to consume just 500 calories~~ a ~~day can almost likely be starving themselves~~. ~~Because~~ the ~~body~~ is a ~~bit more selected to taking in over 2000 calories a day or even more~~, ~~then~~ the ~~abrupt drop~~ of ~~calorie intake~~ can ~~signal your body into starvation mode~~. ~~In this way~~, the ~~body's metabolism will slow right down to save energy~~.<br><br>~~Food items wealthy in fibers: The foods that are very very wealthy in fibers can better basal metabolic rate at~~ a ~~steady pace~~. ~~Actually~~, the ~~fiber content inside the food items would enable to digest or break the food particles at~~ a ~~steady pace~~. ~~Further, you~~ are ~~able to even add protein rich items to improve~~ the ~~body muscles~~. ~~Acai berry~~, ~~prunes plus grapes could furthermore be further added inside~~ the ~~diet chart~~.<br><br>It is ~~simple to see should you need iodine~~. ~~First~~, ~~its~~ a ~~advantageous idea to test oneself using liquid iodine~~. ~~Simply paint~~ a ~~round patch on~~ the ~~stomach and watch to see how long your body takes to absorb~~ it. ~~If it~~ is ~~gone in~~ a ~~few hours~~, ~~then we want to supplement~~.<br><br>~~To eat enough~~, ~~find your Activity level found on~~ the ~~chart~~. ~~Once~~ we ~~have chosen an honest level~~, ~~take~~ a ~~bmr plus Multiply it by your BMR~~. ~~This might give your estimated calorie expenditure~~. ~~Note which this might~~ be ~~only~~ a ~~"ball park" figure plus your actual needs could be more or less depending on your true activity level plus body fat percentage~~. As both are ~~stochastic~~ and ~~adaptive, you will want to frequently revisit your BMR to get an honest and honest amount to effectively eat the correct amount of calories~~.<br><br>~~The true secret to weight loss is a easy 1, although it can~~ be ~~a challenge~~. ~~Just a few little changes that you create inside a every~~-~~day lifestyle usually assist we lose~~ the ~~additional pounds without resorting to the expense~~ (~~both financial and physical~~) of ~~fad diets~~.<br><br>~~To sum everything up the simple formula for losing fat~~ is ~~having~~ the ~~calories burned better than~~ a ~~calories consumed~~. ~~Calculate a estimated RMR then decide~~ on ~~daily calorie consumption.~~		In [[mathematics]], '''limit cardinals''' are certain [[cardinal number]]s. A cardinal number λ is a '''weak limit cardinal''' if λ is neither a [[successor cardinal]] nor zero. This means that one cannot "reach" λ by repeated successor operations. These cardinals are sometimes called simply "limit cardinals" when the context is clear.

			A cardinal λ is a '''strong limit cardinal''' if λ cannot be reached by repeated [[Power set\|powerset]] operations. This means that λ is nonzero and, for all κ < λ, 2<sup>κ</sup> < λ. Every strong limit cardinal is also a weak limit cardinal, because κ<sup>+</sup> ≤ 2<sup>κ</sup> for every cardinal κ, where κ<sup>+</sup> denotes the successor cardinal of κ.

			The first infinite cardinal, <math>\aleph_0</math> ([[Aleph number#Aleph-naught\|aleph-naught]]), is a strong limit cardinal, and hence also a weak limit cardinal.

			== Constructions ==

			One way to construct limit cardinals is via the union operation: <math>\aleph_{\omega}</math> is a weak limit cardinal, defined as the union of all the alephs before it; and in general <math>\aleph_{\lambda}</math> for any [[limit ordinal]] λ is a weak limit cardinal.

			The [[beth number\|ב operation]] can be used to obtain strong limit cardinals. This operation is a map from ordinals to cardinals defined as
			:<math>\beth_{0} = \aleph_0,</math>
			:<math>\beth_{\alpha+1} = 2^{\beth_{\alpha}},</math> (the smallest ordinal [[equinumerous]] with the powerset)
			:If λ is a limit ordinal, <math>\beth_{\lambda} = \bigcup \{ \beth_{\alpha} : \alpha < \lambda\}.</math>
			The cardinal
			:<math>\beth_{\omega} = \bigcup \{ \beth_{0}, \beth_{1}, \beth_{2}, \ldots \} = \bigcup_{n < \omega} \beth_{n} </math>
			is a strong limit cardinal of [[cofinality]] ω. More generally, given any ordinal α, the cardinal
			:<math>\beth_{\alpha+\omega} = \bigcup_{n < \omega} \beth_{\alpha+n} </math>
			is a strong limit cardinal. Thus there are arbitrarily large strong limit cardinals.

			== Relationship with ordinal subscripts ==

			If the [[axiom of choice]] holds, every cardinal number has an [[initial ordinal]]. If that initial ordinal is <math>\omega_{\lambda} \,,</math> then the cardinal number is of the form <math>\aleph_\lambda</math> for the same ordinal subscript λ. The ordinal λ determines whether <math>\aleph_\lambda</math> is a weak limit cardinal. Because <math>\aleph_{\alpha^+} = (\aleph_\alpha)^+ \,,</math> if λ is a successor ordinal then <math>\aleph_\lambda</math> is not a weak limit. Conversely, if a cardinal κ is a successor cardinal, say <math>\kappa = (\aleph_{\alpha})^+ \,,</math> then <math>\kappa = \aleph_{\alpha^+} \,.</math> Thus, in general, <math>\aleph_\lambda</math> is a weak limit cardinal if and only if λ is zero or a limit ordinal.

			Although the ordinal subscript tells whether a cardinal is a weak limit, it does not tell whether a cardinal is a strong limit. For example, [[Zermelo–Fraenkel set theory\|ZFC]] proves that <math>\aleph_\omega</math> is a weak limit cardinal, but neither proves nor disproves that <math>\aleph_\omega</math> is a strong limit cardinal (Hrbacek and Jech 1999:168). The [[generalized continuum hypothesis]] states that <math>\kappa^+ = 2^{\kappa} \,</math> for every infinite cardinal κ. Under this hypothesis, the notions of weak and strong limit cardinals coincide.

			== The notion of inaccessibility and large cardinals ==

			The preceding defines a notion of "inaccessibility": we are dealing with cases where it is no longer enough to do finitely many iterations of the successor and powerset operations; hence the phrase "cannot be reached" in both of the intuitive definitions above. But the "union operation" always provides another way of "accessing" these cardinals (and indeed, such is the case of limit ordinals as well). Stronger notions of inaccessibility can be defined using [[cofinality]]. For a weak (resp. strong) limit cardinal κ the requirement that cf(κ) = κ (i.e. κ be [[regular cardinal\|regular]]) so that κ cannot be expressed as a sum (union) of fewer than κ smaller cardinals. Such a cardinal is called a [[inaccessible cardinal\|weakly (resp. strongly) inaccessible cardinal]]. The preceding examples both are singular cardinals of cofinality ω and hence they are not inaccessible.

			<math>\aleph_0</math> would be an inaccessible cardinal of both "strengths" except that the definition of inaccessible requires that they be uncountable. Standard Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC) cannot even prove the consistency of the existence of an inaccessible cardinal of either kind above <math>\aleph_0</math>, due to [[Gödel's incompleteness theorems\|Gödel's Incompleteness Theorem]]. More specifically, if <math>\kappa</math> is weakly inaccessible then <math>L_{\kappa} \models ZFC</math>. These form the first in a hierarchy of [[large cardinal]]s.

			== See also ==

			* [[Cardinal number]]

			== References ==
			* {{citation
			\| last1= Hrbacek \| first1 = Karel
			\| last2=Jech \| first2=Thomas
			\| title = Introduction to Set Theory
			\| edition = 3
			\| year = 1999 \| isbn = 0-8247-7915-0
			}}
			* {{Citation \| last1=Jech \| first1=Thomas \| author1-link=Thomas Jech \| title=Set Theory \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| edition=third millennium \| series=Springer Monographs in Mathematics \| isbn=978-3-540-44085-7 \| doi=10.1007/3-540-44761-X \| year=2003}}
			* {{Citation \| last1=Kunen \| first1=Kenneth \| author1-link=Kenneth Kunen \| title=Set theory: An introduction to independence proofs \| publisher=[[Elsevier]] \| isbn=978-0-444-86839-8 \| year=1980}}

			==External links==

			* http://www.ii.com/math/cardinals/ Infinite ink on cardinals

			[[Category:Set theory]]
			[[Category:Cardinal numbers]]

71.202.180.170: /* Duality */

2012-08-31T07:37:26Z

Duality

New page

The action of pedaling a bicycle aids in weight reduction. Uphill cycling plus high strength cycling is an great means of developing the muscles in the body to create them stronger. At the same time it burns fat too. You need both to happen and cycling accomplishes this. On the other hand, overweight people who usually [http://safedietplans.com/bmr-calculator bmr calculator] try to consume just 500 calories a day can almost likely be starving themselves. Because the body is a bit more selected to taking in over 2000 calories a day or even more, then the abrupt drop of calorie intake can signal your body into starvation mode. In this way, the body's metabolism will slow right down to save energy. Food items wealthy in fibers: The foods that are very very wealthy in fibers can better basal metabolic rate at a steady pace. Actually, the fiber content inside the food items would enable to digest or break the food particles at a steady pace. Further, you are able to even add protein rich items to improve the body muscles. Acai berry, prunes plus grapes could furthermore be further added inside the diet chart. It is simple to see should you need iodine. First, its a advantageous idea to test oneself using liquid iodine. Simply paint a round patch on the stomach and watch to see how long your body takes to absorb it. If it is gone in a few hours, then we want to supplement. To eat enough, find your Activity level found on the chart. Once we have chosen an honest level, take a bmr plus Multiply it by your BMR. This might give your estimated calorie expenditure. Note which this might be only a "ball park" figure plus your actual needs could be more or less depending on your true activity level plus body fat percentage. As both are stochastic and adaptive, you will want to frequently revisit your BMR to get an honest and honest amount to effectively eat the correct amount of calories. The true secret to weight loss is a easy 1, although it can be a challenge. Just a few little changes that you create inside a every-day lifestyle usually assist we lose the additional pounds without resorting to the expense (both financial and physical) of fad diets. To sum everything up the simple formula for losing fat is having the calories burned better than a calories consumed. Calculate a estimated RMR then decide on daily calorie consumption.