Logarithmic derivative - Revision history

en>Joeinwiki at 14:30, 28 November 2014

2014-11-28T14:30:43Z

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	~~In [[mathematics]], a '''partition''', ''P'' of an [[interval (mathematics)\|interval]] [''a'', ''b''] on the [[real number\|real]] line is a finite [[sequence]] of the form~~		Im addicted to my hobby Skydiving. <br>I to learn German in my spare time.<br><br>Here is my page - [http://Www.ageplay.wiki/index.php?title=User:VicenteHutchins FIFA Coin Generator]

	~~:''a'' = ''x''<sub>0</sub> < ''x''<sub>1</sub> < ''x''<sub>2</sub> <~~ .~~.. < ''x''<sub>''n''~~<~~/sub~~> ~~= ''b''.~~

	~~Another partition of the given interval, ''Q'', is defined as a '''refinement of the partition''', ''P'', when it contains all the points of ''P'' and possibly some other points as well;~~ ~~the partition ''Q'' is said~~ to be “finer” than ''P''. Given two partitions, ''P'' and ''Q'', one can always form their '''common refinement''', denoted ''P'' &or; ''Q'', which consists of all the points of ''P'' and ''Q'', re-numbered in order.<ref>{{cite book\|author=Brannan, D.A.\|title=A First Course in ~~Mathematical Analysis\|publisher=Cambridge University Press\|year=2006\|isbn=9781139458955\|page=262\|url=http://books.google.com/books?id=N8bL9lQUGJgC&pg=PA262}}</ref>~~

	~~The '''norm''' (or '''mesh''') of the partition~~

	~~:''x''<sub>0</sub> < ''x''<sub>1</sub> < ''x''<sub>2</sub> < .~~.~~. < ''x''~~<~~sub~~>~~''n''~~<~~/sub~~>

	is the length of the longest of these subintervals,<ref>{{Cite book\|author=Hijab, Omar\|title=Introduction to Calculus and Classical Analysis\|publisher=Springer\|year=2011\|isbn=9781441994882\|page=60\|url=http://books.google.com/books?id=_gb9fMqur9kC&pg=PA60}}</ref><ref>{{Cite book\|author=Zorich, Vladimir A.\|title=Mathematical Analysis II\|publisher=Springer\|year=2004\|isbn=9783540406334\|page~~=108\|url=http://books.google.com/books?id=XF8W9W~~-~~eyrgC&pg=PA108}}</ref> that is~~

	~~:max{ \|''x''<sub>''i''</sub> − ''x''<sub>''i''−1</sub>\| : ''i'' = 1, ..., ''n'' }.~~

	~~Partitions are used in the theory of the~~ [[Riemann integral]], the [[Riemann–Stieltjes integral]] and the [[regulated integral]]. Specifically, as finer partitions of a given interval are considered, their mesh approaches zero and the [[Riemann sum]] based on a given partition approaches the [[Riemann integral]].<ref>{{cite book\|author=Limaye, Balmohan\|title=A Course in Calculus and Real Analysis\|publisher=Springer\|year=2006\|isbn=9780387364254\|page=213\|url=http://~~books~~.~~google~~.~~com~~/~~books?id=Ou53zXSBdocC&pg=PA213}}</ref>~~

	~~A '''tagged partition'''<ref>{{cite book\|authors=Dudley, Richard M~~. ~~& Norvaiša, Rimas\|title=Concrete Functional Calculus\|publisher=Springer\|year=2010\|isbn=9781441969507\|page=2\|url=http://books.google.com/books~~?id=fuuB59EiIagC&pg=PA2}}</ref> is a partition of a given interval together with a finite sequence of numbers ''t''<sub>0</sub>, ..., ''t''<sub>''n''−1</sub> subject to the conditions that for each ''i'',

	~~:''x''<sub>i</sub> ≤ t<sub>i</sub> ≤ x<sub>i+1</sub>.~~

	In other words, a tagged partition is a partition together with a distinguished point of every subinterval: its mesh is defined in the same way as for an ordinary partition. It is possible to define a [[partial order]] on the set of all tagged partitions by saying that one tagged partition is bigger than another if the bigger one is a refinement of the smaller one.

	Suppose that <math>\scriptstyle x_0,\ldots,x_n</math> together with <math>\scriptstyle t_0,\ldots,t_{n-1}</math> is a tagged partition of <math>[a, b]</math>, and that <math>\scriptstyle y_0,\ldots,y_m</math> together with <math>\scriptstyle s_0,\ldots,s_{m-1}</math> is another tagged partition of <math>[a,b]</math>. We say that <math>\scriptstyle y_0,\ldots,y_m</math> and <math>\scriptstyle s_0,\ldots,s_{m-1}</math> together is a '''refinement of a tagged partition''' <math>\scriptstyle x_0,\ldots,x_n</math> together with <math>\scriptstyle t_0,\ldots,t_{n-1}</math> if for each integer <math>i</math> with <math>\scriptstyle 0 \le i \le n</math>, there is an integer <math>r(i)</math> such that <math>\scriptstyle x_i = y_{r(i)}</math> and such that <math>t_i = s_j</math> for some <math>j</math> with <math>\scriptstyle r(i) \le j \le r(i+1)-1</math>. Said more simply, a refinement of a tagged partition takes the starting partition and adds more tags, but does not take any away.

	~~== See also ==~~
	* [[Regulated integral]]
	* [[Riemann integral]]
	* [[Riemann–Stieltjes integral]]
	* [[Partition of a set]]

	~~== References ==~~
	~~{{Reflist}}~~

	~~==Further reading==~~
	* {{cite book \| last=Gordon \| first=Russell A. \| title=The integrals of Lebesgue, Denjoy, Perron, and [[Ralph Henstock\|Henstock]] \| series=Graduate Studies in Mathematics, 4 \| publisher=American Mathematical Society \| location=Providence, RI \| year=1994 \| isbn=0-8218-3805-9 }}

	~~[[Category~~:~~Mathematical analysis]~~]

en>Ginsuloft: Reverted edits by 39.48.54.247 (talk) to last version by AeronBuchanan

2013-09-29T14:21:43Z

Reverted edits by 39.48.54.247 (talk) to last version by AeronBuchanan

← Older revision		Revision as of 16:21, 29 September 2013
Line 1:		Line 1:
	~~For anyone who is having trouble seeing per game while you are available it~~, ~~try adjusting~~ a ~~new brightness environment. The make the display seem clear~~, ~~enhancing your gaming expertise. And let~~'~~s face it~~, ~~you won~~'~~t achieve any kind connected with success if you not able to see what you~~'~~re doing, so make~~ the ~~casino game meet your needs.~~<br><br>~~If you~~'~~ve got to reload a real arms when playing battle of clans hack that~~'~~s shooting entailed, always capture cover first~~. ~~It genuinely is common for players to be gunned low while a reload is in fact happening, and you see helplessly~~. ~~Do Not even let it happen you! Find somewhere that will conceal before you fire up to reload~~.<br><br>~~Gemstone are known~~ as ~~typically~~ the ~~games primary forex. The Jewels are would purchase resources along via speeding up numerous powerful tasks. The Gemstones can also be would often buy bonus items. Apart from that~~, it ~~can~~ possibly ~~let~~ the ~~leader seen any undesired debris when you want~~ to ~~obtain a additional gems~~. ~~Players~~ can ~~quickly obtain Gems through concluding numerous tasks~~ or ~~perhaps using~~ the ~~clash~~ of ~~clans identify available online~~.<~~br><br~~>A ~~great deal as now, there exists a minimum of social options / capacities with this game i~~. ~~If you liked this article and also you would like to be given more info pertaining to [~~http://~~circuspartypanama~~.com ~~clash~~ of ~~clans unlimited troops] i implore you to visit~~ the ~~page~~. e. ~~there~~ is ~~not any chat~~, ~~attempting~~ to ~~team track of all friends~~, ~~etc but then again we could expect distinct to improve soon considering~~ that ~~Boom Beach continues to be in their Beta Mode.~~<br><br>~~There~~ are ~~a variety~~ of ~~participants what individuals perform Clash of Clans across~~ the ~~world that provides you~~ the ~~chance to allow them to crew up with clans that have been produced by players from different nations around~~ the ~~world and can also keep living competitive towards other clans~~. ~~This will boost the risk for game considerably more interesting~~ as ~~you will locate a great deal~~ of a ~~number of strategies that might be taken by participants~~ and ~~this boosts~~ the ~~unpredictability section~~. ~~Getting the right strategy to win is where the player's skills are tested~~, ~~though the game is simple to play~~ and ~~understand~~.<br><br>~~Be careful about letting your young person play online video games~~, ~~especially games with function sound~~. ~~There can be foul language in these kind channels~~, ~~in addition~~ a ~~lot~~ of ~~bullying behavior~~. ~~There could also be child predators in these types of chat rooms~~. ~~Know what your child is going through and surveil these talk times due~~ to ~~their protection.~~<br><br>~~Contests are some~~ of the ~~finest kinds of amusement around~~. ~~They end up being also probably~~ the the ~~majority pricey types~~ of ~~entertainment~~, with ~~console games~~ and ~~this range from $50 at $60~~, and ~~consoles around their own inside~~ a ~~[https:~~//~~www~~.~~flickr~~.~~com/search/?q~~=~~person%27s person's~~] ~~100s~~. ~~It will be possible to spend considerably on clash~~ of ~~clans hack~~ and ~~console purchases~~, ~~and you can pick out about them by the following paragraphs.~~		In [[mathematics]], a '''partition''', ''P'' of an [[interval (mathematics)\|interval]] [''a'', ''b''] on the [[real number\|real]] line is a finite [[sequence]] of the form

			:''a'' = ''x''<sub>0</sub> < ''x''<sub>1</sub> < ''x''<sub>2</sub> < ... < ''x''<sub>''n''</sub> = ''b''.

			Another partition of the given interval, ''Q'', is defined as a '''refinement of the partition''', ''P'', when it contains all the points of ''P'' and possibly some other points as well; the partition ''Q'' is said to be “finer” than ''P''. Given two partitions, ''P'' and ''Q'', one can always form their '''common refinement''', denoted ''P'' &or; ''Q'', which consists of all the points of ''P'' and ''Q'', re-numbered in order.<ref>{{cite book\|author=Brannan, D.A.\|title=A First Course in Mathematical Analysis\|publisher=Cambridge University Press\|year=2006\|isbn=9781139458955\|page=262\|url=http://books.google.com/books?id=N8bL9lQUGJgC&pg=PA262}}</ref>

			The '''norm''' (or '''mesh''') of the partition

			:''x''<sub>0</sub> < ''x''<sub>1</sub> < ''x''<sub>2</sub> < ... < ''x''<sub>''n''</sub>

			is the length of the longest of these subintervals,<ref>{{Cite book\|author=Hijab, Omar\|title=Introduction to Calculus and Classical Analysis\|publisher=Springer\|year=2011\|isbn=9781441994882\|page=60\|url=http://books.google.com/books?id=_gb9fMqur9kC&pg=PA60}}</ref><ref>{{Cite book\|author=Zorich, Vladimir A.\|title=Mathematical Analysis II\|publisher=Springer\|year=2004\|isbn=9783540406334\|page=108\|url=http://books.google.com/books?id=XF8W9W-eyrgC&pg=PA108}}</ref> that is

			:max{ \|''x''<sub>''i''</sub> − ''x''<sub>''i''−1</sub>\| : ''i'' = 1, ..., ''n'' }.

			Partitions are used in the theory of the [[Riemann integral]], the [[Riemann–Stieltjes integral]] and the [[regulated integral]]. Specifically, as finer partitions of a given interval are considered, their mesh approaches zero and the [[Riemann sum]] based on a given partition approaches the [[Riemann integral]].<ref>{{cite book\|author=Limaye, Balmohan\|title=A Course in Calculus and Real Analysis\|publisher=Springer\|year=2006\|isbn=9780387364254\|page=213\|url=http://books.google.com/books?id=Ou53zXSBdocC&pg=PA213}}</ref>

			A '''tagged partition'''<ref>{{cite book\|authors=Dudley, Richard M. & Norvaiša, Rimas\|title=Concrete Functional Calculus\|publisher=Springer\|year=2010\|isbn=9781441969507\|page=2\|url=http://books.google.com/books?id=fuuB59EiIagC&pg=PA2}}</ref> is a partition of a given interval together with a finite sequence of numbers ''t''<sub>0</sub>, ..., ''t''<sub>''n''−1</sub> subject to the conditions that for each ''i'',

			:''x''<sub>i</sub> ≤ t<sub>i</sub> ≤ x<sub>i+1</sub>.

			In other words, a tagged partition is a partition together with a distinguished point of every subinterval: its mesh is defined in the same way as for an ordinary partition. It is possible to define a [[partial order]] on the set of all tagged partitions by saying that one tagged partition is bigger than another if the bigger one is a refinement of the smaller one.

			Suppose that <math>\scriptstyle x_0,\ldots,x_n</math> together with <math>\scriptstyle t_0,\ldots,t_{n-1}</math> is a tagged partition of <math>[a, b]</math>, and that <math>\scriptstyle y_0,\ldots,y_m</math> together with <math>\scriptstyle s_0,\ldots,s_{m-1}</math> is another tagged partition of <math>[a,b]</math>. We say that <math>\scriptstyle y_0,\ldots,y_m</math> and <math>\scriptstyle s_0,\ldots,s_{m-1}</math> together is a '''refinement of a tagged partition''' <math>\scriptstyle x_0,\ldots,x_n</math> together with <math>\scriptstyle t_0,\ldots,t_{n-1}</math> if for each integer <math>i</math> with <math>\scriptstyle 0 \le i \le n</math>, there is an integer <math>r(i)</math> such that <math>\scriptstyle x_i = y_{r(i)}</math> and such that <math>t_i = s_j</math> for some <math>j</math> with <math>\scriptstyle r(i) \le j \le r(i+1)-1</math>. Said more simply, a refinement of a tagged partition takes the starting partition and adds more tags, but does not take any away.

			== See also ==
			* [[Regulated integral]]
			* [[Riemann integral]]
			* [[Riemann–Stieltjes integral]]
			* [[Partition of a set]]

			== References ==
			{{Reflist}}

			==Further reading==
			* {{cite book \| last=Gordon \| first=Russell A. \| title=The integrals of Lebesgue, Denjoy, Perron, and [[Ralph Henstock\|Henstock]] \| series=Graduate Studies in Mathematics, 4 \| publisher=American Mathematical Society \| location=Providence, RI \| year=1994 \| isbn=0-8218-3805-9 }}

			[[Category:Mathematical analysis]]

en>Csigabi: /* Examples */ +hu

2012-06-10T08:45:45Z

Examples: +hu

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