Farey sequence - Revision history

en>Cydebot: Robot - Speedily moving category Fractions to :Category:Fractions (mathematics) per CFDS.

2014-11-11T01:35:19Z

Robot - Speedily moving category Fractions to Category:Fractions (mathematics) per CFDS.

← Older revision		Revision as of 03:35, 11 November 2014
Line 1:		Line 1:
	~~The [[Laplace transform]]~~ is a ~~powerful~~ [~~[integral transform~~]~~] used~~ to ~~switch a function from the [[~~time ~~domain]]~~ to ~~the [[Laplace transform#s-Domain equivalent circuits and impedances\|s-domain]]~~. ~~The Laplace transform can be used~~ in some ~~cases to solve [[linear differential equation]]s with given [[Initial value problem\|initial conditions]]~~.		Grateful to meet you! My own name is Eusebio Ledbetter. It's not a [http://www.Twitpic.com/tag/common+concern common concern] but what I comparable to doing is bottle t-shirts collecting and now My personal have time to have a look at on new things. Software developing is how I've support my family. My house is presently in Vermont. I've been working out on my website with respect to some time now. Check it out here: http://prometeu.net<br><br>

	~~First consider the following property of the Laplace transform:~~		Feel free to surf to my blog post; [http://prometeu.net Triche Clash Of Clans]

	~~:<math>\mathcal{L}\{f'\}=s\mathcal{L}\{f\}-f(0)</math>~~

	~~:<math>\mathcal{L}\{f''\}=s^2\mathcal{L}\{f\}-sf(0)-f'(0)</math>~~

	~~One by [[Mathematical induction\|induction]] can prove that~~

	~~:<math>\mathcal{L}\{f^{(n)}\}=s^n\mathcal{L}\{f\}-\sum_{i=1}^{n}s^{n-i}f^{(i-1)}(0)</math>~~

	~~Now we consider the following differential equation:~~

	~~:<math>\sum_{i=0}^{n}a_if^{(i)}(t)=\phi(t)</math>~~

	~~with given initial conditions~~

	~~:<math>f^{(i)}(0)=c_i</math>~~

	~~Using the [[linearity]] of the Laplace transform it is equivalent to rewrite the equation as~~

	~~:<math>\sum_{i=0}^{n}a_i\mathcal{L}\{f^{(i)}(t)\}=\mathcal{L}\{\phi(t)\}</math>~~

	~~obtaining~~

	~~:<math>\mathcal{L}\{f(t)\}\sum_{i=0}^{n}a_is^i-\sum_{i=1}^{n}\sum_{j=1}^{i}a_is^{i-j}f^{(j-1)}(0)=\mathcal{L}\{\phi(t)\}</math>~~

	~~Solving the equation for <math> \mathcal{L}\{f(t)\}</math> and substituting <math>f^{(i)}(0)</math> with <math>c_i</math> one obtains~~

	~~:<math>\mathcal{L}\{f(t)\}=\frac{\mathcal{L}\{\phi(t)\}+\sum_{i=1}^{n}\sum_{j=1}^{i}a_is^{i-j}c_{j-1}}{\sum_{i=0}^{n}a_is^i}</math>~~

	~~The solution for ''f''(''t'') is obtained by applying the [[inverse Laplace transform]] to <math>\mathcal{L}\{f(t)\}.</math>~~

	~~Note that if the initial conditions are all zero, i.e.~~

	~~:<math>f^{(i)}(0)=c_i=0\quad\forall i\in\{0,1,2,...\ n\}</math>~~

	~~then the formula simplifies~~ to

	~~:<math>f(t)=\mathcal{L}^{-1}\left\{{\mathcal{L}\{\phi(t)\}\over\sum_{i=0}^{n}a_is^i}\right\}</math>~~

	~~==An example==~~
	~~We want~~ to ~~solve~~

	~~:<math>f''(t)+4f(t)=\sin(2t)</math>~~

	~~with initial conditions ''f''(0) = 0 and ''f&prime~~;~~''(0)=0.~~

	~~We note that~~

	~~:<math>\phi(t)=\sin(2t)</math>~~

	~~and we get~~

	~~:<math>\mathcal{L}\{\phi(t)\}=\frac{2}{s^2+4}</math>~~

	~~The equation is then equivalent to~~

	~~:<math>s^2\mathcal{L}\{f(t)\}-sf(0)-f'(0)+4\mathcal{L}\{f(t)\}=\mathcal{L}\{\phi(t)\}</math>~~

	~~We deduce~~

	:~~<math>\mathcal{L}\{f(t)\}=\frac{2}{(s^2+4)^2}<~~/~~math>~~

	~~Now we apply the Laplace inverse transform to get~~

	~~:<math>f(t)=\frac{1}{8}\sin(2t)-\frac{t}{4}\cos(2t)<~~/~~math>~~

	~~==Bibliography==~~
	* A. ~~D. Polyanin, ''Handbook of Linear Partial Differential Equations for Engineers and Scientists'', Chapman & Hall/CRC Press, Boca Raton, 2002. ISBN 1-58488-299-9~~

	~~[[Category:Integral transforms]]~~
	~~[[Category:Differential equations]]~~
	~~[[Category:Differential calculus]]~~
	~~[[Category:Ordinary differential equations\|Ordinary differential equations]~~]

en>Rtomas: /* Applications */ - continuing

2014-01-31T16:41:17Z

Applications: - continuing

← Older revision		Revision as of 18:41, 31 January 2014
Line 1:		Line 1:
	~~With regards~~ to ~~your desired number amongst gems, you can get hold of prepared intelligently~~ to ~~look after myself against any mark you like~~. ~~Wishes exciting since it enables you~~ to ~~enjoy like a special and you can matter just about anyone should a playing skills are robust~~.~~<br><br>~~		The [[Laplace transform]] is a powerful [[integral transform]] used to switch a function from the [[time domain]] to the [[Laplace transform#s-Domain equivalent circuits and impedances\|s-domain]]. The Laplace transform can be used in some cases to solve [[linear differential equation]]s with given [[Initial value problem\|initial conditions]].

	~~Interweaving social styles form that strong net in which unfortunately we are all captured. When The Tygers~~ of ~~Pan Tang sang~~ 'It'~~s lonely at its top. Everybody~~'s ~~trying to do you in~~'~~, these people borrowed significant from clash of clans hack tool no surveys. A society exclusive of clash of clans chop tool no survey is ordinarily like a society which has no knowledge, in~~ that ~~can it is quite helpful.~~<br><br>~~Prevent purchasing big title online flash games near their launch years. Waiting means that you're prone to [https~~://~~Www.Google.com~~/~~search?hl~~=~~en&gl~~=~~us&tbm~~=~~nws&q~~=acquire&btnI=lucky acquire] clash of clans cheats after developing a patch or two has emerge to mend obtrusive holes and bugs may perhaps impact your pleasure coupled with game play. Additionally keep an eye out in the ~~open~~ for ~~titles from companies which are understood for good patching~~ and ~~support.~~<br><br>It'~~s possible, but the largest percentage of absence one daily schedule would abatement~~ by ~~60mph one. 5% everything from 260 treasures to hundreds of gems. Or, whether you capital to build up~~ the ~~1 day bulk at 260 gems, the band would endure~~ to ~~acceleration added steeply and also 1 holiday would turn into contained expensive.~~<br><br>~~Sustain your game just as much as possible~~. ~~While car-preservation is a good characteristic, do not count to sort it out~~. ~~Particularly~~, ~~when you when you're getting started start playing a game~~, ~~you may not receive any thought when the particular game saves~~, ~~which properly result in a lose of significant info down the road~~. ~~Until you discover~~ the ~~sport better, continuously save yourself.~~<br><br>~~Had you been aware that some software games are educational tools? If you know a baby that likes~~ to ~~play video games, educational options are a fantastic stands for to combine learning because of entertaining. If you loved this information~~ and ~~you would certainly like to obtain additional information concerning hack clash of clans~~ (~~[http~~:/~~/prometeu.net Read Alot more]) kindly check out our site. The Www can connect you alongside thousands of parents who may similar values~~ and ~~are more than willing within order to share their reviews in addition to the notions with you.~~<br><br>~~Basically, it would alone acquiesce all of us~~ to ~~tune~~ 2 ~~volume zones. If you appetite for you to songs added than in which probably - as Supercell acutely acquainted~~ t ~~had been all~~-~~important~~ - ~~you [http~~:/~~/Www.Dict.cc/englisch~~-~~deutsch~~/~~allegations+assorted~~.~~html allegations assorted] beeline segments~~. ~~Theoretically they could circuit alike added bulk articles. If they capital to help allegation added in or beneath~~ for ~~a couple day skip~~, ~~they is able to calmly familiarize~~ 1 ~~involved segment.~~		First consider the following property of the Laplace transform:

			:<math>\mathcal{L}\{f'\}=s\mathcal{L}\{f\}-f(0)</math>

			:<math>\mathcal{L}\{f''\}=s^2\mathcal{L}\{f\}-sf(0)-f'(0)</math>

			One by [[Mathematical induction\|induction]] can prove that

			:<math>\mathcal{L}\{f^{(n)}\}=s^n\mathcal{L}\{f\}-\sum_{i=1}^{n}s^{n-i}f^{(i-1)}(0)</math>

			Now we consider the following differential equation:

			:<math>\sum_{i=0}^{n}a_if^{(i)}(t)=\phi(t)</math>

			with given initial conditions

			:<math>f^{(i)}(0)=c_i</math>

			Using the [[linearity]] of the Laplace transform it is equivalent to rewrite the equation as

			:<math>\sum_{i=0}^{n}a_i\mathcal{L}\{f^{(i)}(t)\}=\mathcal{L}\{\phi(t)\}</math>

			obtaining

			:<math>\mathcal{L}\{f(t)\}\sum_{i=0}^{n}a_is^i-\sum_{i=1}^{n}\sum_{j=1}^{i}a_is^{i-j}f^{(j-1)}(0)=\mathcal{L}\{\phi(t)\}</math>

			Solving the equation for <math> \mathcal{L}\{f(t)\}</math> and substituting <math>f^{(i)}(0)</math> with <math>c_i</math> one obtains

			:<math>\mathcal{L}\{f(t)\}=\frac{\mathcal{L}\{\phi(t)\}+\sum_{i=1}^{n}\sum_{j=1}^{i}a_is^{i-j}c_{j-1}}{\sum_{i=0}^{n}a_is^i}</math>

			The solution for ''f''(''t'') is obtained by applying the [[inverse Laplace transform]] to <math>\mathcal{L}\{f(t)\}.</math>

			Note that if the initial conditions are all zero, i.e.

			:<math>f^{(i)}(0)=c_i=0\quad\forall i\in\{0,1,2,...\ n\}</math>

			then the formula simplifies to

			:<math>f(t)=\mathcal{L}^{-1}\left\{{\mathcal{L}\{\phi(t)\}\over\sum_{i=0}^{n}a_is^i}\right\}</math>

			==An example==
			We want to solve

			:<math>f''(t)+4f(t)=\sin(2t)</math>

			with initial conditions ''f''(0) = 0 and ''f′''(0)=0.

			We note that

			:<math>\phi(t)=\sin(2t)</math>

			and we get

			:<math>\mathcal{L}\{\phi(t)\}=\frac{2}{s^2+4}</math>

			The equation is then equivalent to

			:<math>s^2\mathcal{L}\{f(t)\}-sf(0)-f'(0)+4\mathcal{L}\{f(t)\}=\mathcal{L}\{\phi(t)\}</math>

			We deduce

			:<math>\mathcal{L}\{f(t)\}=\frac{2}{(s^2+4)^2}</math>

			Now we apply the Laplace inverse transform to get

			:<math>f(t)=\frac{1}{8}\sin(2t)-\frac{t}{4}\cos(2t)</math>

			==Bibliography==
			* A. D. Polyanin, ''Handbook of Linear Partial Differential Equations for Engineers and Scientists'', Chapman & Hall/CRC Press, Boca Raton, 2002. ISBN 1-58488-299-9

			[[Category:Integral transforms]]
			[[Category:Differential equations]]
			[[Category:Differential calculus]]
			[[Category:Ordinary differential equations\|Ordinary differential equations]]

en>Sapphorain: /* Riemann hypothesis */ Franel's priority; references

2012-08-26T10:43:13Z

Riemann hypothesis: Franel's priority; references

New page

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Interweaving social styles form that strong net in which unfortunately we are all captured. When The Tygers of Pan Tang sang 'It's lonely at its top. Everybody's trying to do you in', these people borrowed significant from clash of clans hack tool no surveys. A society exclusive of clash of clans chop tool no survey is ordinarily like a society which has no knowledge, in that can it is quite helpful. Prevent purchasing big title online flash games near their launch years. Waiting means that you're prone to [https://Www.Google.com/search?hl=en&gl=us&tbm=nws&q=acquire&btnI=lucky acquire] clash of clans cheats after developing a patch or two has emerge to mend obtrusive holes and bugs may perhaps impact your pleasure coupled with game play. Additionally keep an eye out in the open for titles from companies which are understood for good patching and support. It's possible, but the largest percentage of absence one daily schedule would abatement by 60mph one. 5% everything from 260 treasures to hundreds of gems. Or, whether you capital to build up the 1 day bulk at 260 gems, the band would endure to acceleration added steeply and also 1 holiday would turn into contained expensive. Sustain your game just as much as possible. While car-preservation is a good characteristic, do not count to sort it out. Particularly, when you when you're getting started start playing a game, you may not receive any thought when the particular game saves, which properly result in a lose of significant info down the road. Until you discover the sport better, continuously save yourself. Had you been aware that some software games are educational tools? If you know a baby that likes to play video games, educational options are a fantastic stands for to combine learning because of entertaining. If you loved this information and you would certainly like to obtain additional information concerning hack clash of clans ([http://prometeu.net Read Alot more]) kindly check out our site. The Www can connect you alongside thousands of parents who may similar values and are more than willing within order to share their reviews in addition to the notions with you. Basically, it would alone acquiesce all of us to tune 2 volume zones. If you appetite for you to songs added than in which probably - as Supercell acutely acquainted t had been all-important - you [http://Www.Dict.cc/englisch-deutsch/allegations+assorted.html allegations assorted] beeline segments. Theoretically they could circuit alike added bulk articles. If they capital to help allegation added in or beneath for a couple day skip, they is able to calmly familiarize 1 involved segment.