Graduate Aptitude Test in Engineering - Revision history

en>Engineering Guy: Undid revision 641563054 by 218.248.13.210 (talk)

2015-01-08T21:07:42Z

Undid revision 641563054 by 218.248.13.210 (talk)

← Older revision		Revision as of 23:07, 8 January 2015
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	~~Alyson is the name individuals use~~ to ~~call~~ me and I ~~think it seems quite great when you say it~~. ~~My day job~~ is ~~a travel agent~~. ~~Her family life in Ohio~~. ~~Playing badminton is a thing that~~ he ~~is completely addicted to~~.<br><br>My web ~~page :: telephone~~ psychic ([http://~~kard~~.dk/?p=~~24252 Test by click the "Remove Empty Lines" button and watch these blank lines between this text disappear.~~])		I would like to introduce myself to you, I am Andrew and my spouse doesn't like it at all. What me and my family members love is doing ballet but I've been taking on new things recently. Alaska is where he's usually been residing. Since he was 18 he's been working phone psychic readings ([http://c045.danah.co.kr/home/index.php?document_srl=1356970&mid=qna c045.danah.co.kr]) as an information officer [http://appin.co.kr/board_Zqtv22/688025 tarot readings] but he ideas on changing it.<br><br>Also visit my web-site - online psychic ([http://gcjcteam.org/index.php?mid=etc_video&document_srl=696611&sort_index=regdate&order_type=desc Read More Listed here])

en>Intelligentguy89: /* Latest list of companies (GATE 2014) */

2014-02-24T23:00:06Z

Latest list of companies (GATE 2014)

← Older revision		Revision as of 01:00, 25 February 2014
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	~~In [[mathematics]],~~ the ~~concepts of '''essential supremum''' and '''essential infimum''' are related~~ to ~~the notions of [[supremum]]~~ and ~~[[infimum]], but the former are more relevant in [[measure theory]], where one often deals with statements that are not valid everywhere, that~~ is ~~for ''all'' elements in~~ a ~~[[Set (mathematics)\|set]], but rather ''[[almost everywhere]]'', that is, except on a [[null set\|set of measure zero]]~~.		Alyson is the name individuals use to call me and I think it seems quite great when you say it. My day job is a travel agent. Her family life in Ohio. Playing badminton is a thing that he is completely addicted to.<br><br>My web page :: telephone psychic ([http://kard.dk/?p=24252 Test by click the "Remove Empty Lines" button and watch these blank lines between this text disappear.])

	Let (''X'', Σ, ''μ'') be a [[measure (mathematics)\|measure]] space, and let ''f'' : ''X'' → '''R''' be a [[function (mathematics)\|function]] defined on ''X'' and with [[real number\|real]] values, which is not necessarily [[measurable function\|measurable]]. ~~A real number ''a''~~ is ~~called an ''[[upper bound]]'' for ''f'' if ''f''(''x'') ≤ ''~~a~~'' for all ''x'' in ''X'',~~ that is~~, if the set~~

	~~:<math>\{x\in X: f(x)>a\}</math>~~

	~~is [[empty set\|empty]]~~. ~~In contrast, ''a'' is called an ''essential upper bound'' if the set~~

	:<~~math~~>~~\{x\in X: f(x)>a\}~~<~~/math~~>

	is contained in a set of measure zero, that is to say, if ''f''(''x'') ≤ ''a'' for ''almost all'' ''x'' in ''X''. Then, in the same way as the supremum of ''f'' is defined to be the smallest upper bound, the essential supremum is defined as the smallest essential upper bound.

	~~More formally, the '''essential supremum''' of ''f'', ess sup ''f'', is defined by~~

	:~~<math> \mathrm{ess } \sup f=\inf \{a \in \mathbb{R}~~: ~~\mu~~(~~\{x~~: ~~f(x) > a\}) = 0\}\, <~~/~~math>~~

	~~if the set <math> \{a \in \mathbb{R}: \mu(\{x: f(x) > a\}) = 0\} <~~/~~math> of essential upper bounds is not empty, and ess sup ''f'' = +∞ otherwise~~.

	~~Exactly in the same way one defines the '''essential infimum''' as the largest ''essential lower bound'', that is,~~

	~~:<math> \mathrm{ess } \inf f=\sup \{b \in \mathbb{R}: \mu(\{x: f(x) < b\}) = 0\}\, <~~/~~math>~~

	~~if the set of essential lower bounds is not empty, and as −∞ otherwise.~~

	=~~=Examples==~~

	~~On the real line consider the [[Lebesgue measure]] and its corresponding σ-algebra Σ. Define a function ''f''~~ by the ~~formula~~

	~~:<math> f(x)= \begin{cases} 5, & \text{if } x=1 \\~~
	~~-4, & \text{if } x = -1 \\~~
	~~2, & \text{ otherwise. }~~
	~~\end{cases} </math>~~

	~~The supremum of this function (largest value) is 5,~~ and ~~the infimum (smallest value) is −4. However, the function takes~~ these ~~values only on the sets {1} and {−1} respectively, which are of measure zero. Everywhere else, the function takes the value 2. Thus, the essential supremum and the essential infimum of~~ this ~~function are both 2~~.

	~~As another example, consider the function~~
	~~:<math> f(x)= \begin{cases} x^3, & \text{if } x\in \mathbb Q \\~~
	~~\arctan{x} ,& \text{if } x\in \mathbb R\backslash \mathbb Q \\~~
	~~\end{cases} </math>~~
	~~where '''Q''' denotes the [[rational number~~]]s. This function is unbounded both from above and from below, so its supremum and infimum are ∞ and −∞ respectively. However, from the point of view of the Lebesgue measure, the set of rational numbers is of measure zero; thus, what really matters is what happens in the complement of this set, where the function is given as arctan ''x''. It follows that the essential supremum is π/2 while the essential infimum is −''π''/2.

	~~Lastly, consider the function ''f''(''x'') = ''x''<sup>3</sup> defined for all real ''x''. Its essential supremum is +∞, and its essential infimum is −∞.~~

	~~==Properties==~~
	* <math>\inf f \le \mathrm{ess } \inf f \le \mathrm{ess }\sup f \le \sup f</math>
	* <math>\mathrm{ess }\sup (fg) ~~\le (\mathrm{ess }\sup f)(\mathrm{ess }\sup g)</math> whenever both terms on the right are nonnegative.~~

	~~{{PlanetMath attribution\|id=2044\|title=Essential supremum}}~~

	~~[[Category:Measure theory]]~~
	~~[[Category:Integral calculus]]~~

115.241.91.41: /* Financial Assistance */

2014-02-03T17:39:17Z

Financial Assistance

← Older revision		Revision as of 19:39, 3 February 2014
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	~~Andrew Berryhill is what his wife loves~~ to ~~call him~~ and ~~he totally digs~~ that ~~name~~. ~~My day job~~ is a ~~journey agent~~. ~~The favorite hobby~~ for ~~him~~ and ~~his children~~ is ~~to perform lacross~~ and ~~he would never give it up~~. ~~My spouse~~ and ~~I reside in Kentucky~~.<br><br>~~my web-site~~ ... ~~clairvoyant psychic~~ ([~~http:~~//~~chungmuroresidence~~.~~com~~/xe/~~reservation_branch2~~/~~152663 chungmuroresidence~~.~~com~~])		In [[mathematics]], the concepts of '''essential supremum''' and '''essential infimum''' are related to the notions of [[supremum]] and [[infimum]], but the former are more relevant in [[measure theory]], where one often deals with statements that are not valid everywhere, that is for ''all'' elements in a [[Set (mathematics)\|set]], but rather ''[[almost everywhere]]'', that is, except on a [[null set\|set of measure zero]].

			Let (''X'', Σ, ''μ'') be a [[measure (mathematics)\|measure]] space, and let ''f'' : ''X'' → '''R''' be a [[function (mathematics)\|function]] defined on ''X'' and with [[real number\|real]] values, which is not necessarily [[measurable function\|measurable]]. A real number ''a'' is called an ''[[upper bound]]'' for ''f'' if ''f''(''x'') ≤ ''a'' for all ''x'' in ''X'', that is, if the set

			:<math>\{x\in X: f(x)>a\}</math>

			is [[empty set\|empty]]. In contrast, ''a'' is called an ''essential upper bound'' if the set

			:<math>\{x\in X: f(x)>a\}</math>

			is contained in a set of measure zero, that is to say, if ''f''(''x'') ≤ ''a'' for ''almost all'' ''x'' in ''X''. Then, in the same way as the supremum of ''f'' is defined to be the smallest upper bound, the essential supremum is defined as the smallest essential upper bound.

			More formally, the '''essential supremum''' of ''f'', ess sup ''f'', is defined by

			:<math> \mathrm{ess } \sup f=\inf \{a \in \mathbb{R}: \mu(\{x: f(x) > a\}) = 0\}\, </math>

			if the set <math> \{a \in \mathbb{R}: \mu(\{x: f(x) > a\}) = 0\} </math> of essential upper bounds is not empty, and ess sup ''f'' = +∞ otherwise.

			Exactly in the same way one defines the '''essential infimum''' as the largest ''essential lower bound'', that is,

			:<math> \mathrm{ess } \inf f=\sup \{b \in \mathbb{R}: \mu(\{x: f(x) < b\}) = 0\}\, </math>

			if the set of essential lower bounds is not empty, and as −∞ otherwise.

			==Examples==

			On the real line consider the [[Lebesgue measure]] and its corresponding σ-algebra Σ. Define a function ''f'' by the formula

			:<math> f(x)= \begin{cases} 5, & \text{if } x=1 \\
			-4, & \text{if } x = -1 \\
			2, & \text{ otherwise. }
			\end{cases} </math>

			The supremum of this function (largest value) is 5, and the infimum (smallest value) is −4. However, the function takes these values only on the sets {1} and {−1} respectively, which are of measure zero. Everywhere else, the function takes the value 2. Thus, the essential supremum and the essential infimum of this function are both 2.

			As another example, consider the function
			:<math> f(x)= \begin{cases} x^3, & \text{if } x\in \mathbb Q \\
			\arctan{x} ,& \text{if } x\in \mathbb R\backslash \mathbb Q \\
			\end{cases} </math>
			where '''Q''' denotes the [[rational number]]s. This function is unbounded both from above and from below, so its supremum and infimum are ∞ and −∞ respectively. However, from the point of view of the Lebesgue measure, the set of rational numbers is of measure zero; thus, what really matters is what happens in the complement of this set, where the function is given as arctan ''x''. It follows that the essential supremum is π/2 while the essential infimum is −''π''/2.

			Lastly, consider the function ''f''(''x'') = ''x''<sup>3</sup> defined for all real ''x''. Its essential supremum is +∞, and its essential infimum is −∞.

			==Properties==
			* <math>\inf f \le \mathrm{ess } \inf f \le \mathrm{ess }\sup f \le \sup f</math>
			* <math>\mathrm{ess }\sup (fg) \le (\mathrm{ess }\sup f)(\mathrm{ess }\sup g)</math> whenever both terms on the right are nonnegative.

			{{PlanetMath attribution\|id=2044\|title=Essential supremum}}

			[[Category:Measure theory]]
			[[Category:Integral calculus]]

en>Mahesh Nagre: addes BHEL in PSU's through GATE reference:- http://careers.bhel.in/et_gate2012/static/etgate2012.pdf

2012-09-02T13:58:10Z

addes BHEL in PSU's through GATE reference:- http://careers.bhel.in/et_gate2012/static/etgate2012.pdf

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