Bayesian multivariate linear regression - Revision history

en>Yobot: WP:CHECKWIKI error fixes using AWB (10538)

2014-12-27T08:59:35Z

WP:CHECKWIKI error fixes using AWB (10538)

← Older revision		Revision as of 10:59, 27 December 2014
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	~~{{Unreferenced\|date=December 2009}}~~		Oscar is how he's known as and he totally loves this name. South Dakota is exactly where me and my spouse live. What I adore doing is to gather badges but I've been taking on new things lately. Hiring is her working day job now and she will not change it anytime quickly.<br><br>Here is my page ... [http://www.youporntime.com/user/KBiaggini std testing at home]
	~~[[File:Illustration nested intervals.svg\|thumb\|400px\|right]]~~
	~~In [[mathematics]], a sequence of~~ '~~''nested intervals''' is understood~~ as ~~a collection of sets of real numbers~~

	~~:''I''<sub>''n''</sub>~~

	~~such that each set ''I''<sub>''n''</sub> is an [[interval of the real line]], for ''n'' = 1, 2, 3, ..~~. ~~, and that further~~

	~~:''I''<sub>''n'' + 1 </sub>~~ is ~~a subset of ''I''<sub>''n''</sub>~~

	~~for all ''n''. In other words, the intervals diminish, with the left-hand end moving only towards the right,~~ and ~~the right-hand end only to the left~~.

	~~The main question to be posed is the nature of the [[intersection (set theory)\|intersection]] of all the ''~~I~~''<sub>''n''</sub>. Without any further information, all that can be said~~ is ~~that the intersection ''J'' of all the ''~~I'~~'<sub>''n''</sub>, i.e~~. ~~the set of all points common to the intervals,~~ is ~~either the [[empty set]], a point, or some interval~~.

	~~The possibility of an empty intersection can be illustrated by the intersection when ''I''~~<~~sub~~>~~''n''~~<~~/sub~~> is ~~the [[open interval]]~~
	~~:(0, 2<sup>−''n''</sup>)~~.

	~~Here the intersection is empty, because no number ''x'' is both greater than 0 and less than every fraction 2<sup>−''n''</sup>~~.

	~~The situation is different for [[closed interval]]s~~. ~~The ''nested intervals theorem'' states that if each ''I''<sub>''n''</sub> is a closed and bounded interval, say~~

	~~:''I''<sub>''n''</sub> =~~ [~~''a''<sub>''n''</sub>, ''b''<sub>''n''</sub>]~~

	~~with~~

	:~~''a''<sub>''n''</sub> ≤ ''b''<sub>''n''<~~/~~sub>~~

	~~then under the assumption of nesting, the intersection of the ''I''<sub>''n''<~~/~~sub> is not empty~~. ~~It may be a singleton set {''c''}, or another closed interval [''a'', ''b'']. More explicitly, the requirement of nesting means that~~

	~~: ''a''<sub>''n''</sub> ≤ ''a''<sub>''n'' + 1</sub>~~

	~~and~~

	~~: ''b''<sub>''n''</sub> ≥ ''b''<sub>''n'' + 1</sub>~~.

	~~Moreover, if the length of the intervals converges to 0, then the intersection of the ''I''<sub>''n''<~~/~~sub> is a singleton.~~

	~~One can consider the complement of each interval, written as <math>(-\infty,a_n) \cup (b_n, \infty)<~~/math>. By [[De Morgan's laws]], the complement of the intersection is a union of two disjoint open sets. By the [[connectedness]] of the [[real line]] there must be something between them. This shows that the intersection of (even an [[uncountable]] number of) nested, closed, and bounded intervals is nonempty.

	~~==Higher dimensions==~~
	In two dimensions there is a similar result: nested [[closed disk]]s in the plane must have a common intersection. This result was shown by [[Hermann Weyl]] to classify the singular behaviour of certain [[differential equation]]s.

	~~==See also==~~
	*[[Bisection]]
	*[[Cantor's Intersection Theorem]]

	~~{{DEFAULTSORT:Nested Intervals}}~~

	~~[[Category:Sets of real numbers]]~~
	~~[[Category:Theorems in real analysis]~~]

83.252.203.32: /* Details */ link to the moore-penrose pseudoinverse article

2013-10-12T13:42:33Z

Details: link to the moore-penrose pseudoinverse article

← Older revision		Revision as of 15:42, 12 October 2013
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	~~Greetings!~~ I ~~am Myrtle Shroyer~~. ~~North Dakota is exactly where me~~ and ~~my husband live. Managing people~~ is ~~what~~ I do and the ~~salary has been truly satisfying~~. The ~~preferred hobby for my kids and me~~ is ~~to perform baseball but~~ I ~~haven~~'~~t produced~~ a ~~dime with it~~.<br><br>~~my web site~~ ... ~~std home test -~~ [~~http~~://~~immbooks~~.~~com~~/~~blogs~~/~~post~~/~~44599 Recommended Internet page~~],		{{Unreferenced\|date=December 2009}}
			[[File:Illustration nested intervals.svg\|thumb\|400px\|right]]
			In [[mathematics]], a sequence of '''nested intervals''' is understood as a collection of sets of real numbers

			:''I''<sub>''n''</sub>

			such that each set ''I''<sub>''n''</sub> is an [[interval of the real line]], for ''n'' = 1, 2, 3, ... , and that further

			:''I''<sub>''n'' + 1 </sub> is a subset of ''I''<sub>''n''</sub>

			for all ''n''. In other words, the intervals diminish, with the left-hand end moving only towards the right, and the right-hand end only to the left.

			The main question to be posed is the nature of the [[intersection (set theory)\|intersection]] of all the ''I''<sub>''n''</sub>. Without any further information, all that can be said is that the intersection ''J'' of all the ''I''<sub>''n''</sub>, i.e. the set of all points common to the intervals, is either the [[empty set]], a point, or some interval.

			The possibility of an empty intersection can be illustrated by the intersection when ''I''<sub>''n''</sub> is the [[open interval]]
			:(0, 2<sup>−''n''</sup>).

			Here the intersection is empty, because no number ''x'' is both greater than 0 and less than every fraction 2<sup>−''n''</sup>.

			The situation is different for [[closed interval]]s. The ''nested intervals theorem'' states that if each ''I''<sub>''n''</sub> is a closed and bounded interval, say

			:''I''<sub>''n''</sub> = [''a''<sub>''n''</sub>, ''b''<sub>''n''</sub>]

			with

			:''a''<sub>''n''</sub> ≤ ''b''<sub>''n''</sub>

			then under the assumption of nesting, the intersection of the ''I''<sub>''n''</sub> is not empty. It may be a singleton set {''c''}, or another closed interval [''a'', ''b'']. More explicitly, the requirement of nesting means that

			: ''a''<sub>''n''</sub> ≤ ''a''<sub>''n'' + 1</sub>

			and

			: ''b''<sub>''n''</sub> ≥ ''b''<sub>''n'' + 1</sub>.

			Moreover, if the length of the intervals converges to 0, then the intersection of the ''I''<sub>''n''</sub> is a singleton.

			One can consider the complement of each interval, written as <math>(-\infty,a_n) \cup (b_n, \infty)</math>. By [[De Morgan's laws]], the complement of the intersection is a union of two disjoint open sets. By the [[connectedness]] of the [[real line]] there must be something between them. This shows that the intersection of (even an [[uncountable]] number of) nested, closed, and bounded intervals is nonempty.

			==Higher dimensions==
			In two dimensions there is a similar result: nested [[closed disk]]s in the plane must have a common intersection. This result was shown by [[Hermann Weyl]] to classify the singular behaviour of certain [[differential equation]]s.

			==See also==
			*[[Bisection]]
			*[[Cantor's Intersection Theorem]]

			{{DEFAULTSORT:Nested Intervals}}

			[[Category:Sets of real numbers]]
			[[Category:Theorems in real analysis]]

en>Andrewaskew: Fixed order of categorisation.

2012-07-03T01:53:24Z

Fixed order of categorisation.

New page

Greetings! I am Myrtle Shroyer. North Dakota is exactly where me and my husband live. Managing people is what I do and the salary has been truly satisfying. The preferred hobby for my kids and me is to perform baseball but I haven't produced a dime with it.<br><br>my web site ... std home test - [http://immbooks.com/blogs/post/44599 Recommended Internet page],