From 450da746bc7d886274ad2d2b6426b4168e5906db Mon Sep 17 00:00:00 2001 From: Sasserisop Date: Mon, 4 Dec 2023 23:21:10 -0700 Subject: [PATCH] revised fourier and added lec 34 --- content/Fourier Series (lec 28-29).md | 211 +-- ...ial differential equations (lec 30-34).md} | 71 +- ...bles & Eigen value problems (lec 26-28).md | 7 +- content/_index.md | 4 +- .../Drawing-2023-11-22-13.15.26.excalidraw.md | 1158 +++++++++++------ ...Drawing-2023-11-22-13.15.26.excalidraw.png | Bin 25876 -> 24965 bytes content/drawings/fouriercosineofsin.png | Bin 0 -> 105509 bytes 7 files changed, 961 insertions(+), 490 deletions(-) rename content/{Partial differential equations (lec 30-33).md => Partial differential equations (lec 30-34).md} (76%) create mode 100644 content/drawings/fouriercosineofsin.png diff --git a/content/Fourier Series (lec 28-29).md b/content/Fourier Series (lec 28-29).md index 1660d8b..5556766 100644 --- a/content/Fourier Series (lec 28-29).md +++ b/content/Fourier Series (lec 28-29).md @@ -1,154 +1,177 @@ - +#fourier Remember the heat flow equation? We obtained that it's solution could be expressed in the form: $$\sum_{n=1}^\infty c_{n}\sin\left( \frac{n\pi x}{L} \right)\quad\text{for}\quad0\leq x\leq L$$ -But what is $c_{n}$? They are the coefficients of a fourier transform. We want to develop a way to compute them. -Let's derive how to compute the coefficients of a fourier transform. (feel free to skip to the end) -$f(x)=\sum_{n=1}^\infty b_{n}\sin\left( \frac{n\pi x}{L} \right)$ where L is length of the rod +But what is $c_{n}$? They are the coefficients of a Fourier transform. We want to develop a way to compute them. +Let's derive how to compute the coefficients of a Fourier transform. (feel free to skip to the end) +$f(x)=\sum_{n=1}^\infty b_{n}\sin\left( \frac{n\pi x}{L} \right)$ where $L$ is length of the rod -This is a Fourier series, it's a more general form of what we have above: +This is a Fourier series: it's a more general form of what we have above. $f(x)=\frac{a_{0}}{2}+\sum_{n=1}^\infty\left( a_{n}\cos\left( \frac{n\pi x}{L} \right) + b_{n}\sin\left( \frac{n\pi x}{L}\right) \right)$ $x \in [-L,L]$ -almost everywhere piecewise continuous (?) -has a lot of benefits over taylor series. $f(x)$ doesn't have to be infinitely differentiable (analytic) -f(x) can even have jump discontinuities -lets assume the equation is true when $x \in [-L,L]$ +It converges to $f(x)$ almost everywhere (convergence will be discussed below) +Has a lot of benefits over Taylor series. $f(x)$ doesn't have to be infinitely differentiable (analytic) +$f(x)$ can even have jump discontinuities +Let's assume the equation is true when $x \in [-L,L]$ +Integrate both sides, it will tell us the DC offset: $\int _{-L} ^L f(x) \, dx=\int _{-L}^L \frac{a_{0}}{2} \, dx+\int _{-L}^L (\text{put summation here}) \, dx$ $\int _{-L}^L \cos\left( \frac{n\pi x}{L} \right) \, dx=\frac{L}{n\pi}\sin\left( \frac{n\pi x}{L} \right)|_{-L}^L=0$ same for $\int _{-L}^L \sin\left( \frac{n\pi x}{L} \right)\, dx=0$ (it equals 0) so $\int _{-L} ^L f(x) \, dx=\int _{-L}^L \frac{a_{0}}{2} \, dx+\int _{-L}^L0 \, dx$ -$\int _{-L} ^L f(x) \, dx=\int _{-L}^L \frac{a_{0}}{2} \, dx+\int _{-L}^L0 \, dx$ -$a_{0}=\frac{1}{L}\int _{L}^{2?a_{0}L} f(x) \, dx$ -$\int _{-L}^L f(x)\cos\left( \frac{m\pi x}{L} \right)\, dx=\frac{a_{0}}{2}\cancelto{ 0 }{ \int _{-L}^L \cos\left( \frac{m\pi x}{L} \right) \, dx }+\sum_{n=1}^\infty\left( a_{n}\int _{-L}^L\cos\left( \frac{n\pi x}{L} \right)\cos\left( \frac{m\pi x}{L} \right) \, dx \right)+b_{n}\int _{-L}^L \sin\left( \frac{n\pi x}{L} \right)\cos\left( \frac{m\pi x}{L} \right) \, dx$ +$\int _{-L} ^L f(x) \, dx=a_{0}L$ +$a_{0}=\frac{1}{L}\int _{-L}^{L} f(x) \, dx$ +Now let's multiply both sides by $\cos\left( \frac{m\pi x}{L} \right)$ and integrate both sides, this will tell us the $\cos$ components: +$\int _{-L}^L f(x)\cos\left( \frac{m\pi x}{L} \right)\, dx=\frac{a_{0}}{2}\cancelto{ 0 }{ \int _{-L}^L \cos\left( \frac{m\pi x}{L} \right) \, dx }+\sum_{n=1}^\infty\left( a_{n}\int _{-L}^L\cos\left( \frac{n\pi x}{L} \right)\cos\left( \frac{m\pi x}{L} \right) \, dx +b_{n}\int _{-L}^L \sin\left( \frac{n\pi x}{L} \right)\cos\left( \frac{m\pi x}{L} \right)\right) \, dx$ use trig identities (will be provided on exam): $\cos(\alpha)\cos(\beta)=\frac{1}{2}(\cos(\alpha-\beta)+\cos(\alpha+\beta))$ $\sin(\alpha)\cos(\beta)=\frac{1}{2}(\sin(\alpha+\beta)+\sin(\alpha-\beta))$ $\sin(\alpha)\sin(\beta)=\frac{1}{2}(\cos(\alpha-\beta)-\cos(\alpha+\beta))$ -$\int _{-L}^L \cos \frac{n\pi x}{L}\cos \frac{m\pi x}{L}\, dx=\frac{1}{2}(\int _{-L}^L \left( \cos \frac{(n-m)\pi x}{L} \right) \, dx+\cancelto{ 0 }{ \frac{\cos(n+m)\pi x}{L} }dx$ +$\int _{-L}^L \cos \frac{n\pi x}{L}\cos \frac{m\pi x}{L}\, dx=\frac{1}{2}(\int _{-L}^L \left( \cos(\frac{(n-m)\pi x}{L} )+\cancelto{ 0 }{ \cos(\frac{(n+m)\pi x}{L} })\right)dx$ $= \begin{cases}0 & n\ne m \\L & n=m\end{cases}$ - -$\int _{-L}^L \sin \frac{n\pi x}{L}\cos \frac{m\pi}{L} \, dx=0;$ -$\int _{-L}^L \sin \frac{n\pi x}{L}\cos \frac{m\pi}{L} \, dx=\begin{cases}0 & n\ne m \\L & n=m\end{cases}$ - -going back, +$\int _{-L}^L \sin \frac{n\pi x}{L}\cos \frac{m\pi x}{L} \, dx=\int _{-L}^L \text{odd}\, dx=0$ +so: +$\int _{-L} ^L f(x)\cos\left( \frac{m\pi x}{L} \right)\, dx=a_{m}L$ +Similarly can be done for when multiplying both sides by $\sin\left( \frac{m\pi x}{L} \right)$ and integrating both sides to find the $\sin$ coefficients: +$\int _{-L}^L f(x)\sin\left( \frac{m\pi x}{L} \right)\, dx=\frac{a_{0}}{2}\cancelto{ 0 }{ \int _{-L}^L \sin\left( \frac{m\pi x}{L} \right) \, dx }+\sum_{n=1}^\infty\left( a_{n}\cancelto{ \text{odd} }{ \int _{-L}^L\cos\left( \frac{n\pi x}{L} \right)\sin\left( \frac{m\pi x}{L} \right) } \, dx +b_{n}\int _{-L}^L \sin\left( \frac{n\pi x}{L} \right)\sin\left( \frac{m\pi x}{L} \right)\right) \, dx$ +$\int _{-L} ^L \sin\left( \frac{n\pi x}{L} \right)\sin\left( \frac{m\pi x}{L} \right) \, dx=\frac{1}{2}\int_{-L}^L \cos\left(\frac{(n-m)\pi x}{L} \right)-\cos\left( \frac{(n+m)\pi x}{L} \right)dx$ +$=\begin{cases}0, & n\ne m \\L, & n=m\end{cases}$ +so: +$\int _{-L} ^L f(x)\sin\left( \frac{m\pi x}{L} \right)\, dx=b_{m}L$ +In conclusion: $$a_{m}=\frac{1}{L}\int _{-L}^L f(x)\cos \frac{m\pi x}{L} \, dx \quad\text{valid for all }m=0,1,2,\dots$$ $$b_{m}=\frac{1}{L}\int _{-L}^L f(x)\sin \frac{m\pi x}{L} \, dx=b_{m} \quad m=1,2,\dots$$ -now we know how to compute the coefficients for Fourier series! +Now we know how to compute the coefficients for Fourier series! properties: for functions $f$, $g$, If $\int _{-L}^Lf(x)g(x) \, dx=\begin{cases}0 & f\ne g \\L & f=g \end{cases}$ then $f, g$ are orthogonal -the forier expantion is called an ortho normal expansion, taylor is not ortho normal. +the Fourier expansion is called an ortho normal expansion, Taylor is not orthonormal. #end of lec 28 #start of lec 29 -last lecture we derived how to find the coefficients in a fourier series. +Last lecture we derived how to find the coefficients in a Fourier series. $f(x)=\frac{a_{0}}{2}+\sum_{n=1}^\infty\left( a_{n}\cos\left( \frac{n\pi x}{L} \right) + b_{n}\sin\left( \frac{n\pi x}{L}\right) \right)$ $x \in [-L,L]$ -### Theorem: -If $f$ and $f'$ are piecewise continuous on $[-L,L]$, then the fourier series converges to: +### 1st convergence theorem: +If $f$ and $f'$ are piecewise continuous on $[-L,L]$, then the Fourier series converges to: $\frac{1}{2}(f(x^-)+f(x^+))$ for all $x \in (-L,L)$ -Basically meaning, the fourier series converges. -At $x=\pm L$ the fourier series converges to $\frac{1}{2}(f(-L^+)+f(L^-))$ +and on $x=\pm L$ the Fourier series converges to $\frac{1}{2}(f(-L^+)+f(L^-))$ ![draw](drawings/Drawing-2023-11-22-13.15.26.excalidraw.png) -### Theorem: -If f(x) is continuous on $(-\infty,\infty)$ and $2L$ periodic and if $f'$ is continuous, then the taylor series converges to $f(x)$ everywhere +Recall the definition of piecewise continuous: $f(t)$ is piecewise continuous on an interval $I$ if $f(t)$ is continuous on $I$, except possibly at a finite number of points of jump discontinuity (horizontal asymptotes not allowed). +### 2nd Convergence theorem (uniform convergence): +If $f(x)$ is continuous on $(-\infty,\infty)$ and $2L$ periodic and if $f'$ is piecewise continuous on $[-L,L]$, then its Fourier series converges to $f(x)$ everywhere (i.e., the Fourier series converges uniformly). ![draw](drawings/Drawing-2023-11-22-13.14.05.excalidraw.png) -#ex lets compute the fourier transform of: -$f(x)=\begin{cases}1, & -\pi\leq x\leq 0 \\x, & 0Don't be a silly goose and try changing the bounds by removing that 2 in the front. If you did, you'd also have to change $\sin(x)$ to $\bar{f}$ which is $abs(\sin(x))$ and then you're integrating $a_{n}=\frac{1}{\pi}\int _{-\pi}^\pi |\sin(x)|\cos(nx)\, dx$ which is even$\times$even. -$=\frac{2}{\pi} \frac{1}{2}\int _{0}^\pi (\sin(1-n)x+\sin(n+1)x)\, dx$ -integrating gives you: +Use trig identity: (by the way the identities will be provided in the final exam.) +$=\frac{2}{\pi} \frac{1}{2}\int _{0}^\pi \left[\sin((1-n)x)+\sin((n+1)x)\right]\, dx$ +Integrating gives you: +$\frac{1}{\pi}( \frac{-1}{1-n}\underbrace{ \cos((1-n)x)|_{0}^\pi }_{ (-1)^{n+1}-1 } +\frac{-1}{n+1}\underbrace{ \cos((n+1)x)|_{0}^\pi }_{ (-1)^{n+1}-1 })$ +$a_{n}=-\frac{1}{\pi} \frac{1}{n+1}(-1)^{n+1}+\frac{1}{\pi} \frac{1}{n+1}+\frac{1}{\pi} \frac{1}{-(1-n)}(-1)^{n+1}+\frac{1}{\pi} \frac{1}{1-n}$ $a_{n}=-\frac{1}{\pi} \frac{1}{n+1} (-1)^{n+1}+\frac{1}{\pi} \frac{1}{n+1}+\frac{1}{\pi} \frac{1}{n-1}(-1)^{n-1}-\frac{1}{\pi} \frac{1}{n-1}$ -what about when n=0 or n=1? -$a_{0}=\frac{4}{\pi}=\frac{2}{\pi}\int _{0}^\pi \sin(x) \, dx$ +Assuming that $n\ne0,1$. (note: $n=-1$ is a non-issue since negative coefficients are never considered when taking a Fourier transform.) +So what is $a_{0}, a_{1}$? +$a_{0}=\frac{2}{\pi}\int _{0}^\pi \sin(x) \, dx=\frac{4}{\pi}$ $a_{1}=\frac{2}{\pi}\int _{0}^\pi \sin(x)\cos(x) \, dx=\frac{1}{\pi}\int _{0}^\pi \sin(2x)\, dx=0$ -"0 is a very very special number it took humanity many numbers of years to invent 0" referring to when dividing by 0. -additionally we know that the terms cancel when: +"zero is a very very special number it took humanity many numbers of years to invent zero" referring to when dividing by 0. +Additionally we know that the terms cancel when: $a_{2k-1}=0$ for $k=1,2,\dots$ -$a_{2k}=\frac{2}{\pi} \frac{1}{2k+1}+\frac{2}{\pi} \frac{1}{2k-1}$ for $k=1,2,\dots$ +$a_{2k}=\frac{2}{\pi} \frac{1}{2k+1}-\frac{2}{\pi} \frac{1}{2k-1}$ for $k=1,2,\dots$ then: -$$\bar{f}(x)=\frac{2}{\pi}+\frac{2}{\pi}\sum_{k=1}^\infty\left( \frac{1}{2k+1}+\frac{1}{2k-1} \right)\cos(2kx)$$ - +$$\bar{f}(x)=\frac{2}{\pi}+\frac{2}{\pi}\sum_{k=1}^\infty\left( \frac{1}{2k+1}-\frac{1}{2k-1} \right)\cos(2k\pi x)$$ +Even with 10 terms, we get a pretty good approximation: +![fouriercosineofsin.png](drawings/fouriercosineofsin.png) We have prepared ourselves now, now we start solving PDE's. He's encouraging us to attend the lectures in these last two weeks. He's making it sound like PDE's are hard. \ No newline at end of file diff --git a/content/Partial differential equations (lec 30-33).md b/content/Partial differential equations (lec 30-34).md similarity index 76% rename from content/Partial differential equations (lec 30-33).md rename to content/Partial differential equations (lec 30-34).md index af66de4..ca4056e 100644 --- a/content/Partial differential equations (lec 30-33).md +++ b/content/Partial differential equations (lec 30-34).md @@ -231,9 +231,74 @@ you complaining that your exams are hard theyre not hard. I'm talking about 40 y now we consider a guitar string: ![draw](drawings/Drawing-2023-12-01-13.49.58.excalidraw.png) assuming the thickness of the string is much smaller than the length of the string, which is true. -$\frac{ \partial u^2 }{ \partial t^2 }=\alpha^2 \frac{ \partial^2 u }{ \partial x^2 } \quad 0\leq x\leq L, t>0$ -^ Reminds me of the wave equation from phys 130. +$\frac{ \partial u^2 }{ \partial t^2 }=\alpha^2 \frac{ \partial^2 u }{ \partial x^2 } \quad 0\leq x\leq L,\quad t>0$ +^ Reminds me of the wave equation from Phys 130. +Since the string is tied down on the ends we have the following initial conditions: $u(t,0)=u(t,L)=0 \qquad t>0$ $u(0,x)=f(x)$ $0\leq x\leq L$ $\frac{ \partial u }{ \partial t }(0,x)=g(x)$ $0\leq x\leq L$ -#end of lec 33 \ No newline at end of file +^IBVP of the system. +#end of lec 33 +#start of lec 34 +The wave equation follows many phenomena in electrical engineering. +separation of variables: +$u(t,x)=X(x)T(t)$ +plug in to equation: +$T''X=\alpha^2TX''$ +$\frac{T''}{\alpha^2T}=\frac{X''}{X}=-\lambda$ +^ #evp ! +consider the $X$ side: +$X''+\lambda X=0, \quad X(0)=X(L)=0$ +We've solved this before. +the only non-trivial solutions for that Eigen value problem is: +$\lambda_{n}=(\frac{n\pi}{L})^2$ +$X_{n}(x)=\sin\left( \frac{n\pi x}{L} \right)$ for $n=1,2,3,\dots$ +$\frac{T''}{\alpha^2T}=-\left( \frac{n\pi}{L} \right)^2$ +$T_{n}''+\left( \frac{\alpha n\pi}{L} \right)^2T_{n}=0$ +characteristic equation: +$r^2+\left( \frac{\alpha n\pi}{L} \right)^2=0$ +$r_{1,2}=\pm i \frac{\alpha n\pi}{L}$ +"Don't memorize the steps. If you try to memorize you will mess up the final for sure. Ask yourself, why am I doing this here?" +$T_{n}(t)=b_{n}\cos\left( \frac{\alpha n\pi}{L}t \right)+a_{n}\sin\left( \frac{\alpha n\pi}{L}t \right)$ +$u_{n}(t,x)=\left( b_{n}\cos\left( \frac{\alpha n\pi}{L} \right)+a_{n}\sin\left( \frac{\alpha n\pi}{L}t \right) \right)\sin\left( \frac{n\pi x}{L} \right)$ +if you sum all these modes, you get the solution: +$$u_{n}(t,x)=\sum_{n=1}^\infty\left( b_{n}\cos\left( \frac{\alpha n\pi}{L} \right)+a_{n}\sin\left( \frac{\alpha n\pi}{L}t \right) \right)\sin\left( \frac{n\pi x}{L} \right)$$ + +"mathematics and reality do align very well, if the speed of the string is different the solution differs aswell." +$u(0,x)=f(x)=\sum_{n=1}^\infty b_{n}\sin\left( \frac{n\pi x}{L} \right)$ +^ that's starting to look familiar. +$\implies b_{n}=\frac{2}{L}\int _{0}^L f(x)\sin\left( \frac{n\pi x}{L} \right) \, dx$ +where does it converge? well $f(x)$ and $f'(x)$ are both continuous, so it converges everywhere. +$\frac{ \partial u }{ \partial t }(0,x)=g(x)=\sum_{n=1}^\infty\underbrace{ a_{n} \frac{\alpha n\pi}{L} }_{ }\sin\left( \frac{\alpha n\pi}{L} \right)$ +$a_{n} \frac{\alpha n\pi}{L}$ are the Fourier $\sin$ coefficients of $g(x)$ +$a_{n}=\frac{2}{\alpha n\pi}\int _{0}^L g(x)\sin\left( \frac{n\pi x}{L} \right)\, dx$ +$\alpha^2$ is the Hooke modulus of the string btw. +remember the heat equation, the amplitude is exponentionally decreasing, +here the amplitude is oscillatory and doesnt increae in time. boi-oi-oi-oing +to make it more releasitic we have to add a term for resistance, and we end up with a b in the characteristic equation for T. +btw this equation models the electromagnetic feild, to some approximation. +the lowest mode is called the fundemental mode, the following terms after are called harmonics. +If two instruments play the same note (same fundemental frequency), they still sound different! and that's because of the difference in harmonics. +The modes are standing waves in the string. +"my claim is that any object, including social objects ,can be described by waves. Everything is a wave." +you can model elementary particle behaviours with solitons (non linear waves.) +in life in the real world, all waves have finite speed. +So thats why its important to learn the wave equation. its the prototype to waves. +"waves are the fundamental object. [...]. So that's why it's important, these are the fundamental objects of nature here." +$f(x)=\begin{cases}x, & 0\leq x\leq \frac{\pi}{2}\\ \ \pi-x, & \frac{\pi}{2}I'm gonna be honest, I'm lost in this derivation already. What is $Q(t,x)$? +$Q(t,x)$ is a source term, a term to model internally produced heat. +>I'm gonna be honest, I'm lost in this derivation already. divide by $a\Delta x\Delta t$ $C(x)\rho(x)\frac{(u(t+\Delta t,x)-u(t,x))}{\Delta t}=\frac{ k(x+\Delta x) \frac{\partial u}{\partial x}(t,x+\Delta x)-k(x) \frac{\partial u}{\partial x}(t,x)}{\nabla x}+Q(t,x)$ @@ -32,7 +33,7 @@ thermodynamics can be very important for electrical engineers, for instance the ## Separation of variables & Eigen value problems #ex #SoV #evp (This is more of a case study than an example.) -Rewrite the equation we derived by grouping the constant terms into one constant $D$ +Rewrite the equation we derived by grouping the constant terms into one constant $D$ and assume for this problem that the internally generated heat is $0$ (i.e., $Q(x,t)=0$) $\frac{ \partial u }{ \partial t }=D \frac{ \partial^2 u }{ \partial x^2 }, \quad 0\leq x\leq L, \quad t>0$ boundary conditions: $u(t,0)=u(t,L)=0 , \quad t>0$ (simple case) @@ -116,7 +117,7 @@ lets go back to the problem and focus on $T$: $\frac{T'}{DT}=\frac{X''}{X}=-\lambda$ $\frac{T'}{T}=-\left( \frac{n\pi}{L} \right)^2D$ this is a separable equation. -We can treat the function T as a variable: +We can treat the function $T$ as a variable: $\frac{dT}{dt} \frac{1}{T}=-\left( \frac{n\pi}{L} \right)^2D$ $\int{dT} \frac{1}{T}=\int-\left( \frac{n\pi}{L} \right)^2Ddt$ $\ln(T)=-\left( \frac{n\pi}{L} \right)Dt+c_{n}$ diff --git a/content/_index.md b/content/_index.md index e516061..080b50d 100644 --- a/content/_index.md +++ b/content/_index.md @@ -24,8 +24,8 @@ I have written these notes for myself, I thought it would be cool to share them. [Systems of linear equations (lec 21-22)](systems-of-linear-equations-lec-21-22.html) [Power series (lec 22-25)](power-series-lec-22-25.html) [Separation of variables & Eigen value problems (lec 26-28)](separation-of-variables-eigen-value-problems-lec-26-28.html) -[Fourier series (lec 28-29)](fourier-series-lec-28-29.html) (raw notes, not reviewed or revised yet.) -[Partial differential equations (lec 30-33)](partial-differential-equations-lec-30-33.html) (raw notes, not reviewed or revised yet.) +[Fourier series (lec 28-29)](fourier-series-lec-28-29.html) +[Partial differential equations (lec 30-34)](partial-differential-equations-lec-30-34.html) (raw notes, not reviewed or revised yet.)
[How to solve any DE, a flow chart](Solve-any-DE.png) (Last updated Oct 1st, needs revision. But it gives a nice overview.) [Big LT table (.png)](drawings/bigLTtable.png) diff --git a/content/drawings/Drawing-2023-11-22-13.15.26.excalidraw.md b/content/drawings/Drawing-2023-11-22-13.15.26.excalidraw.md index 5fc5fcc..28da041 100644 --- a/content/drawings/Drawing-2023-11-22-13.15.26.excalidraw.md +++ b/content/drawings/Drawing-2023-11-22-13.15.26.excalidraw.md @@ -17,780 +17,1162 @@ ZMzYBXUtiRoIACyhQ4zZAHoFAc0JRJQgEYA6bGwC2CgF7N6hbEcK4OCtptbErHAL RY8RMpWdx8Q1TdIEfARcZgRmBShcZQUARm0ATm1YhJo6IIR9BA4oZm4AbXAwUDBS -iBJuCAB5fX0OAHV6ADMADgBVABUjOAArAFkAYQAGKoAhAA1lAE000shYREqmwIRP +6HhxdAAzQIRPKn4yxhZ2LjQAFgBmRshm1k4AOU4xbnaADjGEzq6ABjGeiEJmABEM -Kn4yzG5nAGYdoe0eABYANgB2E4BWWJ3z2Jazy43IGG2W+L2eS54z2JOhk5HI6XHb +qDruKoIwhZJuCAB5fX0OAHV6KrGAVQAVIzgAKwBZAGEZg4AhAA1lAE000qQKqEfD -PCAUEjqbgnc7JHj/BItE47HjvL5gyQIQjKaTcY7aQGPHgpM7wy4ndFFSDWZTBbhD +4ADKsGC+0kuGwGkCALKzCgpDYAGsEKcSOpuLEFsjURiITAoRJBB5EZBUX5JBxwnk -MHMKCkNgAawQAzY+DYpEqAGJYghBYLZmVNLhsKzlCyhBxiJzubyJMzrMw4LhAjlR 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