Random permutation statistics - Revision history

en>Zahlentheorie: /* External links: keys and boxes */

2014-07-30T21:17:53Z

External links: keys and boxes

← Older revision		Revision as of 23:17, 30 July 2014
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	~~Let me first begin by introducing myself. My~~ name is ~~Boyd Butts although it~~ is ~~not the name on~~ my ~~beginning certification~~. ~~Body building is what my~~ family ~~and~~ I ~~appreciate. South Dakota~~ is ~~exactly where I've always been living. For many years~~ I'~~ve been operating as~~ a ~~payroll clerk~~.<br><br>~~Also visit my webpage home std test [~~[http://~~torontocartridge~~.com/~~uncategorized~~/~~the~~-~~ideal-way-to-battle-a-yeast-infection~~/ ~~linked resource site]~~]		Nice to meet you, my name is Refugia. Hiring is my profession. Years ago he moved to North Dakota and his family enjoys it. What I love performing is playing baseball but I haven't produced a dime with it.<br><br>My page; [http://btcsoc.com/index.php?do=/profile-4621/info/ btcsoc.com]

en>BG19bot: WP:CHECKWIKI error fix for #99. Broken sup tag. Do general fixes if a problem exists. -, replaced: ^{''m'' → ^''m'' using AWB (9957)}

2014-03-01T08:46:34Z

WP:CHECKWIKI error fix for #99. Broken sup tag. Do general fixes if a problem exists. -, replaced: ''m'' → ''m'' using AWB (9957)

← Older revision		Revision as of 10:46, 1 March 2014
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	~~In theory of [[oscillation\|vibration]]s, '''Duhamel's integral''' is a way of calculating the response of [[linear system]]s and [[structures]] to arbitrary time-varying external [[excitation]]s.~~		Let me first begin by introducing myself. My name is Boyd Butts although it is not the name on my beginning certification. Body building is what my family and I appreciate. South Dakota is exactly where I've always been living. For many years I've been operating as a payroll clerk.<br><br>Also visit my webpage home std test [[http://torontocartridge.com/uncategorized/the-ideal-way-to-battle-a-yeast-infection/ linked resource site]]

	~~==Introduction==~~
	~~===Background===~~
	~~The response of a linear, viscously damped [[single-degree of freedom]] (SDOF) system to a time-varying mechanical excitation ''p''(''t'') is given~~ by ~~the following second-order [[ordinary differential equation]]~~
	~~:<math>m\frac{{d^2 x(t)}}{{dt^2 }} + c\frac{{dx(t)}}{{dt}} + kx(t) = p(t)</math>~~
	~~where ''m'' is the (equivalent) mass, ''x'' stands for the amplitude of vibration, ''t'' for time, ''c'' for the viscous damping coefficient, and ''k'' for the [[stiffness]] of the system or structure~~.

	~~If a system~~ is ~~initially rest at its [[Mechanical equilibrium\|equilibrium]] position, from where~~ it is ~~acted upon by a unit-impulse at~~ the ~~instance ''t''=0, i~~.e.~~, ''p''(''t'') in the equation above~~ is a [[Dirac delta function]] ''δ''(''t''), <math>x(0) = \left. {\frac{{dx}}{{dt}}} \right\|_{t = 0} = 0</math>, then by solving the differential equation one can get a [[fundamental solution]] (known as a '''unit-impulse response function''')
	~~:<math>h(t)=\begin{cases} \frac{1}{{m\omega _d }}e^{ - \varsigma \omega _n t} \sin \omega _d t, & t > 0 \\ 0, & t < 0 \end{cases}</math>~~
	where <math>\varsigma = \frac{c}{2\sqrt{k m}}</math> is called the [[damping ratio]] of the system, <math>\omega _n=\sqrt{\frac{k}{m}}</math> is the natural [[angular frequency]] of the undamped system (when ''c''=0) and <math>\omega _d = \omega _n \sqrt {1 - \varsigma ^2 } </math> is the [[circular frequency]] when damping effect is taken into account (when <math>c \ne 0</math>). If the impulse happens at ''t''=''τ'' instead of ''t''=0, i.~~e. <math>p(t)=\delta (t - \tau )</math>, the impulse response is~~
	~~:<math>h(t - \tau ) = \frac{1}{{m\omega _d }}e^{ - \varsigma \omega _n (t - \tau )} \sin [\omega _d (t - \tau )]</math>，<math>t \ge \tau </math>~~

	~~===Conclusion===~~
	~~Regarding the arbitrarily varying excitation ''p''(~~'~~'t'')~~ as a ~~[[Superposition principle\|superposition]] of a series of impulses:~~
	~~:<math>p(t) \approx \sum {p(\tau ) \cdot \Delta \tau \cdot \delta } (t - \tau )</math>~~
	~~then it is known from the linearity of system that the overall response can also be broken down into the superposition of a series of impulse-responses:~~
	~~:<math>x(t) \approx \sum {p(\tau ) \cdot \Delta \tau \cdot h} (t - \tau )</math>~~
	~~Letting <math>\Delta \tau \to 0</math>, and replacing the summation by [[Integral\|integration]], the above equation is strictly valid~~
	~~:<math>x(t) = \int_0^t {p(\tau )h(t - \tau )d\tau } </math>~~
	~~Substituting the expression of ''h''(''t''-''τ'') into the above equation leads to the general expression of Duhamel's integral~~
	~~:<math>x(t) = \frac{1}{{m\omega _d }}\int_0^t {p(\tau )e^{ - \varsigma \omega _n (t - \tau )} \sin [\omega _d (t - \tau )]d\tau }</math>~~

	~~===Mathematical Proof===~~
	~~The above SDOF dynamic equilibrium equation in the case ''p(t)=0'' is the [[homogeneous differential equation\|homogeneous equation]]:~~
	~~:<math>\frac{{d^2 x(t)}}{{dt^2 }} + \bar{c}\frac{{dx(t)}}{{dt}} + \bar{k}x(t) = 0</math>, where <math>\bar{c}=\frac{c}{m},\bar{k}=\frac{k}{m} </math>~~
	~~The solution of this equation is:~~
	~~:<math>x_h(t) = C_1~~.~~e^{ -\frac{1}{2}.(\bar{c}+\sqrt{\bar{c}^2-4.\bar{k}}).t}+C_2.e^{ \frac{1}{2}.(-\bar{c}+\sqrt{\bar{c}^2-4.\bar{k}}).t}~~<~~/math~~>
	~~The substitution:~~ <~~math~~>~~A = \frac{1}{2}.(\bar{c}-\sqrt{\bar{c}^2-4.\bar{k}}), \; B=\frac{1}{2}.(\bar{c}+\sqrt{\bar{c}^2-4.\bar{k}}), \; P=\sqrt{\bar{c}^2-4.\bar{k}}, \; P=B-A</math> leads to:~~
	~~:<math>x_h(t) = C_1.e^{ -B.t} \; + \; C_2.e^{ -A.t}</math>~~
	One partial solution of the non-homogeneous equation: <math> \frac{{d^2 x(t)}}{{dt^2 }} + \bar{c}\frac{{dx(t)}}{{dt}} + \bar{k}x(t) = \bar{p(t)}</math>, where <math>\bar{p(t)}=\frac{p(t)}{m}</math>, could be obtained by the Lagrangian method for deriving partial solution of non-homogeneous [[~~ordinary differential equations]].~~

	~~This solution has the form:~~
	~~:<math>x_p(t) = \frac{\int{\bar{p(t)}.e^{At}dt}.e^{-At}-\int{\bar{p(t)}.e^{Bt}dt}.e^{-Bt}}{P}</math>~~
	~~Now substituting~~:~~<math>\int{\bar{p(t)}.e^{At}dt}\|_{t=z}=Q_z, \int{\bar{p(t)}.e^{Bt}dt}\|_{t=z}=R_z <~~/~~math>,where <math> \int{x(t)dt}\|_{t=z} </math> is the [[antiderivative\|primitive]] of ''x(t)'' computed at ''t=z'', in the case ''z=t'' this integral is the primitive itself, yields:~~
	~~:<math>x_p(t) = \frac{Q_t.e^{-At}-R_t.e^{-Bt}}{P}<~~/~~math>~~
	~~Finally the general solution of the above non-homogeneous equation is represented as:~~
	~~:<math>x(t)=x_h(t)+x_p(t)=C_1.e^{ -B.t}+C_2~~.~~e^{ -A.t} +\frac{Q_t.e^{-At}-R_t.e^{-Bt}}{P}<~~/~~math>~~
	~~with time derivative:~~
	~~:<math> \frac{dx}{dt}=-A.e^{-At}.C_2-B.e^{-Bt}.C_1+\frac{1}{P}.[\dot{Q_t}.e^{-At}-A.Q_t.e^{-At}-\dot{R_t}.e^{-Bt}+B.R_t.e^{-Bt}]<~~/~~math>, where <math>\dot{Q_t}=p(t).e^{At},\dot{R_t}=p(t).e^{Bt}</math>~~
	~~In order to find~~ the ~~unknown constants <math>C_1, C_2</math>, zero initial conditions will be applied:~~
	~~:<math>x(t)\|_{t=0} = 0: C_1+C_2+\frac{Q_0.1~~-~~R_0.1}{P}=0</math> ⇒ <math>C_1+C_2=\frac{R_0~~-~~Q_0}{P}</math>~~
	~~:<math>\left. {\frac{{dx}}{{dt}}} \right\|_{t=0} = 0: -A.C_2-B.C_1+\frac{1}{P}.[-A.Q_0+B.R_0]=0</math> ⇒ <math>A.C_2+B.C_1=\frac{1}{P}.[B.R_0-A.Q_0]</math>~~
	~~Now combining both initial conditions together, the next system of equations is observed:~~
	~~:<math>\left.{\begin{alignat}{5}~~
	~~C_1 &&\; + &&\; C_2 &&\; = &&\; \frac{R_0-Q_0}{P} & \\~~
	~~B.C_1 &&\; + &&\; A.C_2 &&\; = &&\; \frac{1}{P}.[B.R_0-A.Q_0]\end{alignat}} \right\|{\begin{alignat}{5}~~
	~~C_1 &&\; = &&\; \frac{R_0}{P} & \\~~
	~~C_2 &&\; = &&\; -\frac{Q_0}{P}\end{alignat}}</math>~~
	~~The back substitution of the constants <math> C_1 </math> and <math> C_2 </math> into the above expression for ''x(t)'' yields:~~
	~~:<math>x(t)=\frac{Q_t-Q_0}{P}.e^{ -A.t}-\frac{R_t-R_0}{P}.e^{ -B.t}</math>~~
	Replacing <math>Q_t-Q_0</math> and <math>R_t-R_0</math> (the difference between the primitives at ''t=t'' and ''t=0'') with [[integral\|definite integrals]] (by another variable ''τ'') will reveal the general solution with zero initial conditions, namely:
	~~:<math>x(t)=\frac{1}{P}.[\int_0^t{\bar{p(\tau)}.e^{A\tau}d\tau}.e^{-At}-\int_0^t{\bar{p(\tau)}.e^{B\tau}d\tau}.e^{-Bt}]</math>~~
	~~Finally substituting <math> c=2.\xi.\omega.m, \; k=\omega^2.m</math>, accordingly <math> \bar{c}=2.\xi.\omega, \bar{k}=\omega^2</math>, where <u>''ξ<1''</u> yields:~~
	~~:<math>P=2.\omega_D.i, \; A=\xi.\omega~~-~~\omega_D.i, \; B=\xi.\omega+\omega_D.i</math>, where <math>\omega_D=\omega.\sqrt{1-\xi^2}</math> and '''''i''''' is the [[imaginary unit]].~~
	~~Substituting this expressions into the above general solution with zero initial conditions and using the [[Euler's formula\|Euler's exponential formula]] will lead~~ to ~~canceling out the imaginary terms and reveals the Duhamel's solution:~~
	~~:<math>x(t)=\frac{1}{\omega_D}\int_0^t{\bar{p(\tau)}e^{~~-~~\xi\omega(t~~-~~\tau)}sin(\omega_D(t~~-~~\tau))d\tau}</math>~~

	~~== See also ==~~
	*[[Duhamel's principle]]

	~~==References==~~
	* Ni Zhenhua, ''Mechanics of Vibrations'', Xi'an Jiaotong University Press, Xi'an, 1990 (in Chinese)
	* R. W. Clough, J. Penzien, ''Dynamics of Structures'', Mc-~~Graw Hill Inc., New York, 1975.~~
	* Anil K. Chopra, ''Dynamics of Structures - Theory and applications to Earthquake Engineering'', Pearson Education Asia Limited and Tsinghua University Press, Beijing, 2001
	* Leonard Meirovitch, ''Elements of Vibration Analysis'', Mc-Graw Hill Inc., Singapore, 1986

	~~==External links==~~
	*[http://tosio.math.toronto.edu/wiki/index.php/~~Duhamel's_formula Duhamel's formula] at "Dispersive Wiki".~~

	~~[[Category:Mechanics]]~~
	~~[[Category:Structural analysis]]~~
	~~[[Category:Integrals~~]]

en>Zahlentheorie: /* Number of permutations containing m cycles: ogf vs. egf */

2014-01-17T04:50:26Z

Number of permutations containing m cycles: ogf vs. egf

← Older revision		Revision as of 06:50, 17 January 2014
Line 1:		Line 1:
	The ~~writer~~ is known as ~~Irwin Wunder but~~ it's ~~not~~ the ~~most masucline name out there~~. One of the ~~things she enjoys most~~ is to ~~do aerobics~~ and ~~now she is trying to make cash~~ with it. ~~California is our beginning location~~. ~~My day job~~ is ~~a meter reader~~.<br><br>~~Here is my site~~ ... [http://~~www~~.~~pinaydiaries~~.~~com~~/~~blog~~/~~104745~~ at ~~home std testing~~]		In theory of [[oscillation\|vibration]]s, '''Duhamel's integral''' is a way of calculating the response of [[linear system]]s and [[structures]] to arbitrary time-varying external [[excitation]]s.

			==Introduction==
			===Background===
			The response of a linear, viscously damped [[single-degree of freedom]] (SDOF) system to a time-varying mechanical excitation ''p''(''t'') is given by the following second-order [[ordinary differential equation]]
			:<math>m\frac{{d^2 x(t)}}{{dt^2 }} + c\frac{{dx(t)}}{{dt}} + kx(t) = p(t)</math>
			where ''m'' is the (equivalent) mass, ''x'' stands for the amplitude of vibration, ''t'' for time, ''c'' for the viscous damping coefficient, and ''k'' for the [[stiffness]] of the system or structure.

			If a system is initially rest at its [[Mechanical equilibrium\|equilibrium]] position, from where it is acted upon by a unit-impulse at the instance ''t''=0, i.e., ''p''(''t'') in the equation above is a [[Dirac delta function]] ''δ''(''t''), <math>x(0) = \left. {\frac{{dx}}{{dt}}} \right\|_{t = 0} = 0</math>, then by solving the differential equation one can get a [[fundamental solution]] (known as a '''unit-impulse response function''')
			:<math>h(t)=\begin{cases} \frac{1}{{m\omega _d }}e^{ - \varsigma \omega _n t} \sin \omega _d t, & t > 0 \\ 0, & t < 0 \end{cases}</math>
			where <math>\varsigma = \frac{c}{2\sqrt{k m}}</math> is called the [[damping ratio]] of the system, <math>\omega _n=\sqrt{\frac{k}{m}}</math> is the natural [[angular frequency]] of the undamped system (when ''c''=0) and <math>\omega _d = \omega _n \sqrt {1 - \varsigma ^2 } </math> is the [[circular frequency]] when damping effect is taken into account (when <math>c \ne 0</math>). If the impulse happens at ''t''=''τ'' instead of ''t''=0, i.e. <math>p(t)=\delta (t - \tau )</math>, the impulse response is
			:<math>h(t - \tau ) = \frac{1}{{m\omega _d }}e^{ - \varsigma \omega _n (t - \tau )} \sin [\omega _d (t - \tau )]</math>，<math>t \ge \tau </math>

			===Conclusion===
			Regarding the arbitrarily varying excitation ''p''(''t'') as a [[Superposition principle\|superposition]] of a series of impulses:
			:<math>p(t) \approx \sum {p(\tau ) \cdot \Delta \tau \cdot \delta } (t - \tau )</math>
			then it is known from the linearity of system that the overall response can also be broken down into the superposition of a series of impulse-responses:
			:<math>x(t) \approx \sum {p(\tau ) \cdot \Delta \tau \cdot h} (t - \tau )</math>
			Letting <math>\Delta \tau \to 0</math>, and replacing the summation by [[Integral\|integration]], the above equation is strictly valid
			:<math>x(t) = \int_0^t {p(\tau )h(t - \tau )d\tau } </math>
			Substituting the expression of ''h''(''t''-''τ'') into the above equation leads to the general expression of Duhamel's integral
			:<math>x(t) = \frac{1}{{m\omega _d }}\int_0^t {p(\tau )e^{ - \varsigma \omega _n (t - \tau )} \sin [\omega _d (t - \tau )]d\tau }</math>

			===Mathematical Proof===
			The above SDOF dynamic equilibrium equation in the case ''p(t)=0'' is the [[homogeneous differential equation\|homogeneous equation]]:
			:<math>\frac{{d^2 x(t)}}{{dt^2 }} + \bar{c}\frac{{dx(t)}}{{dt}} + \bar{k}x(t) = 0</math>, where <math>\bar{c}=\frac{c}{m},\bar{k}=\frac{k}{m} </math>
			The solution of this equation is:
			:<math>x_h(t) = C_1.e^{ -\frac{1}{2}.(\bar{c}+\sqrt{\bar{c}^2-4.\bar{k}}).t}+C_2.e^{ \frac{1}{2}.(-\bar{c}+\sqrt{\bar{c}^2-4.\bar{k}}).t}</math>
			The substitution: <math>A = \frac{1}{2}.(\bar{c}-\sqrt{\bar{c}^2-4.\bar{k}}), \; B=\frac{1}{2}.(\bar{c}+\sqrt{\bar{c}^2-4.\bar{k}}), \; P=\sqrt{\bar{c}^2-4.\bar{k}}, \; P=B-A</math> leads to:
			:<math>x_h(t) = C_1.e^{ -B.t} \; + \; C_2.e^{ -A.t}</math>
			One partial solution of the non-homogeneous equation: <math> \frac{{d^2 x(t)}}{{dt^2 }} + \bar{c}\frac{{dx(t)}}{{dt}} + \bar{k}x(t) = \bar{p(t)}</math>, where <math>\bar{p(t)}=\frac{p(t)}{m}</math>, could be obtained by the Lagrangian method for deriving partial solution of non-homogeneous [[ordinary differential equations]].

			This solution has the form:
			:<math>x_p(t) = \frac{\int{\bar{p(t)}.e^{At}dt}.e^{-At}-\int{\bar{p(t)}.e^{Bt}dt}.e^{-Bt}}{P}</math>
			Now substituting:<math>\int{\bar{p(t)}.e^{At}dt}\|_{t=z}=Q_z, \int{\bar{p(t)}.e^{Bt}dt}\|_{t=z}=R_z </math>,where <math> \int{x(t)dt}\|_{t=z} </math> is the [[antiderivative\|primitive]] of ''x(t)'' computed at ''t=z'', in the case ''z=t'' this integral is the primitive itself, yields:
			:<math>x_p(t) = \frac{Q_t.e^{-At}-R_t.e^{-Bt}}{P}</math>
			Finally the general solution of the above non-homogeneous equation is represented as:
			:<math>x(t)=x_h(t)+x_p(t)=C_1.e^{ -B.t}+C_2.e^{ -A.t} +\frac{Q_t.e^{-At}-R_t.e^{-Bt}}{P}</math>
			with time derivative:
			:<math> \frac{dx}{dt}=-A.e^{-At}.C_2-B.e^{-Bt}.C_1+\frac{1}{P}.[\dot{Q_t}.e^{-At}-A.Q_t.e^{-At}-\dot{R_t}.e^{-Bt}+B.R_t.e^{-Bt}]</math>, where <math>\dot{Q_t}=p(t).e^{At},\dot{R_t}=p(t).e^{Bt}</math>
			In order to find the unknown constants <math>C_1, C_2</math>, zero initial conditions will be applied:
			:<math>x(t)\|_{t=0} = 0: C_1+C_2+\frac{Q_0.1-R_0.1}{P}=0</math> ⇒ <math>C_1+C_2=\frac{R_0-Q_0}{P}</math>
			:<math>\left. {\frac{{dx}}{{dt}}} \right\|_{t=0} = 0: -A.C_2-B.C_1+\frac{1}{P}.[-A.Q_0+B.R_0]=0</math> ⇒ <math>A.C_2+B.C_1=\frac{1}{P}.[B.R_0-A.Q_0]</math>
			Now combining both initial conditions together, the next system of equations is observed:
			:<math>\left.{\begin{alignat}{5}
			C_1 &&\; + &&\; C_2 &&\; = &&\; \frac{R_0-Q_0}{P} & \\
			B.C_1 &&\; + &&\; A.C_2 &&\; = &&\; \frac{1}{P}.[B.R_0-A.Q_0]\end{alignat}} \right\|{\begin{alignat}{5}
			C_1 &&\; = &&\; \frac{R_0}{P} & \\
			C_2 &&\; = &&\; -\frac{Q_0}{P}\end{alignat}}</math>
			The back substitution of the constants <math> C_1 </math> and <math> C_2 </math> into the above expression for ''x(t)'' yields:
			:<math>x(t)=\frac{Q_t-Q_0}{P}.e^{ -A.t}-\frac{R_t-R_0}{P}.e^{ -B.t}</math>
			Replacing <math>Q_t-Q_0</math> and <math>R_t-R_0</math> (the difference between the primitives at ''t=t'' and ''t=0'') with [[integral\|definite integrals]] (by another variable ''τ'') will reveal the general solution with zero initial conditions, namely:
			:<math>x(t)=\frac{1}{P}.[\int_0^t{\bar{p(\tau)}.e^{A\tau}d\tau}.e^{-At}-\int_0^t{\bar{p(\tau)}.e^{B\tau}d\tau}.e^{-Bt}]</math>
			Finally substituting <math> c=2.\xi.\omega.m, \; k=\omega^2.m</math>, accordingly <math> \bar{c}=2.\xi.\omega, \bar{k}=\omega^2</math>, where <u>''ξ<1''</u> yields:
			:<math>P=2.\omega_D.i, \; A=\xi.\omega-\omega_D.i, \; B=\xi.\omega+\omega_D.i</math>, where <math>\omega_D=\omega.\sqrt{1-\xi^2}</math> and '''''i''''' is the [[imaginary unit]].
			Substituting this expressions into the above general solution with zero initial conditions and using the [[Euler's formula\|Euler's exponential formula]] will lead to canceling out the imaginary terms and reveals the Duhamel's solution:
			:<math>x(t)=\frac{1}{\omega_D}\int_0^t{\bar{p(\tau)}e^{-\xi\omega(t-\tau)}sin(\omega_D(t-\tau))d\tau}</math>

			== See also ==
			*[[Duhamel's principle]]

			==References==
			* Ni Zhenhua, ''Mechanics of Vibrations'', Xi'an Jiaotong University Press, Xi'an, 1990 (in Chinese)
			* R. W. Clough, J. Penzien, ''Dynamics of Structures'', Mc-Graw Hill Inc., New York, 1975.
			* Anil K. Chopra, ''Dynamics of Structures - Theory and applications to Earthquake Engineering'', Pearson Education Asia Limited and Tsinghua University Press, Beijing, 2001
			* Leonard Meirovitch, ''Elements of Vibration Analysis'', Mc-Graw Hill Inc., Singapore, 1986

			==External links==
			*[http://tosio.math.toronto.edu/wiki/index.php/Duhamel's_formula Duhamel's formula] at "Dispersive Wiki".

			[[Category:Mechanics]]
			[[Category:Structural analysis]]
			[[Category:Integrals]]

en>Phil Boswell: convert dodgy URL to ID using AWB

2012-08-29T19:55:42Z

convert dodgy URL to ID using AWB

New page

The writer is known as Irwin Wunder but it's not the most masucline name out there. One of the things she enjoys most is to do aerobics and now she is trying to make cash with it. California is our beginning location. My day job is a meter reader. Here is my site ... [http://www.pinaydiaries.com/blog/104745 at home std testing]

Random permutation statistics - Revision history

en>Zahlentheorie: /* External links: keys and boxes */

en>BG19bot: WP:CHECKWIKI error fix for #99. Broken sup tag. Do general fixes if a problem exists. -, replaced: ''m'' → ''m'' using AWB (9957)

en>Zahlentheorie: /* Number of permutations containing m cycles: ogf vs. egf */

en>Phil Boswell: convert dodgy URL to ID using AWB

en>BG19bot: WP:CHECKWIKI error fix for #99. Broken sup tag. Do general fixes if a problem exists. -, replaced: ^{''m'' → ^''m'' using AWB (9957)}