forked from Sasserisop/MATH201
revised Separation of variables & Eigen value problems
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@ -23,7 +23,7 @@ $u'(t-a)=\delta(t-a)$
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What is $\mathcal{L}\{\delta(t-a)\}$?
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What is $\mathcal{L}\{\delta(t-a)\}$?
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$\mathcal{L}\{\delta(t-a)\}=\int _{0} ^\infty \delta(t-a)e^{-st} \, dt$ for $a>0$
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$\mathcal{L}\{\delta(t-a)\}=\int _{0} ^\infty \delta(t-a)e^{-st} \, dt$ for $a>0$
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Using the definition earlier:
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Using the definition of Dirak $\int _{-\infty} ^{\infty} \delta(t-a)f(t)\, dt=f(a)$:
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$$\mathcal{L}\{\delta(t-a)\}=\int _{-\infty} ^{\infty} e^{-st} \delta(t-a) \, dt=e^{-as}$$
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$$\mathcal{L}\{\delta(t-a)\}=\int _{-\infty} ^{\infty} e^{-st} \delta(t-a) \, dt=e^{-as}$$
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## Examples of DE's with Dirak
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## Examples of DE's with Dirak
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#ex #second_order_nonhomogenous #dirak_delta #IVP
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#ex #second_order_nonhomogenous #dirak_delta #IVP
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@ -49,10 +49,10 @@ If $f$ and $f'$ are piecewise continuous on $[-L,L]$, then the fourier series co
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$\frac{1}{2}(f(x^-)+f(x^+))$ for all $x \in (-L,L)$
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$\frac{1}{2}(f(x^-)+f(x^+))$ for all $x \in (-L,L)$
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Basically meaning, the fourier series converges.
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Basically meaning, the fourier series converges.
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At $x=\pm L$ the fourier series converges to $\frac{1}{2}(f(-L^+)+f(L^-))$
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At $x=\pm L$ the fourier series converges to $\frac{1}{2}(f(-L^+)+f(L^-))$
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![[Partial differential equations (lec 26-27) 2023-11-22 13.15.26.excalidraw]]
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### Theorem:
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### Theorem:
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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
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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
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![[Partial differential equations (lec 26-27) 2023-11-22 13.14.05.excalidraw]]
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#ex lets compute the fourier transform of:
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#ex lets compute the fourier transform of:
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$f(x)=\begin{cases}1, & -\pi\leq x\leq 0 \\x, & 0<x\leq \pi\end{cases}$
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$f(x)=\begin{cases}1, & -\pi\leq x\leq 0 \\x, & 0<x\leq \pi\end{cases}$
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$L$ here is $\pi$ clearly.
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$L$ here is $\pi$ clearly.
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@ -124,7 +124,7 @@ then $\frac{a_{0}}{2}+\sum_{n=1}^\infty a_{n}\cos\left( \frac{n\pi x}{L} \right)
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#ex Fourier sine series for $f(x)=x^2$ from $0\leq x\leq \pi$
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#ex Fourier sine series for $f(x)=x^2$ from $0\leq x\leq \pi$
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well that means we want the odd extension:
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well that means we want the odd extension:
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![[Lec 30 2023-11-24 13.15.17.excalidraw]]
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the $\cos()$ ($a_{n}$) terms are zero.
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the $\cos()$ ($a_{n}$) terms are zero.
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the b terms are:
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the b terms are:
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$b_{n}=\frac{2}{\pi}\int _{0}^\pi x^2\sin(nx) \, dx$
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$b_{n}=\frac{2}{\pi}\int _{0}^\pi x^2\sin(nx) \, dx$
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@ -134,7 +134,7 @@ $b_{n}=-\frac{2}{n\pi}\left[ \pi^2(-1)^n-\frac{2}{n^2}\cos(nx)|_{0}^\pi \right]=
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for $n=1,2,3,\dots$ note no $n=0$ so no divison by zero problems here.
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for $n=1,2,3,\dots$ note no $n=0$ so no divison by zero problems here.
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#ex fourier cosine series of $f(x)=\sin(x)$ for $0\leq x\leq \pi$
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#ex fourier cosine series of $f(x)=\sin(x)$ for $0\leq x\leq \pi$
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![[Lec 30 2023-11-24 13.23.08.excalidraw]]
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$a_{n}=\frac{2}{\pi}\int _{0}^\pi \sin(x)\cos(nx)\, dx$ for $n=0,1,2,\dots$
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$a_{n}=\frac{2}{\pi}\int _{0}^\pi \sin(x)\cos(nx)\, dx$ for $n=0,1,2,\dots$
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use trig identity: (by the way the identites will be provided in the exam.)
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use trig identity: (by the way the identites will be provided in the exam.)
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@ -35,7 +35,7 @@ $k=1,l=-3$
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so $x=u+1 \quad y=v-3$
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so $x=u+1 \quad y=v-3$
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$(-3u+v)du+(u+v)dv=0$ //Beautiful! It's homogenous now
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$(-3u+v)du+(u+v)dv=0$ //Beautiful! It's homogenous now
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$\frac{ dv }{ du }=\frac{{3u-v}}{u+v}$
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$\frac{ dv }{ du }=\frac{{3u-v}}{u+v}$
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divide top and bottom by u so we turn the homogenous equation into the form #de_h_type1 and solve it using the tools we developed from lecture 2.
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divide top and bottom by $u$ so we turn the homogenous equation into the form #de_h_type1 and solve it using the tools we developed from lecture 2.
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$\frac{ dv }{ du }=\frac{{3-\frac{v}{u}}}{1+\frac{v}{u}}$
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$\frac{ dv }{ du }=\frac{{3-\frac{v}{u}}}{1+\frac{v}{u}}$
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$\frac{v}{u}=w \quad v=uw \quad \frac{ dv }{ du }=w+u\frac{ dw }{ du }$
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$\frac{v}{u}=w \quad v=uw \quad \frac{ dv }{ du }=w+u\frac{ dw }{ du }$
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@ -8,7 +8,7 @@ $u(t,0)=u(t,L)=0, \quad t>0$
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$u(0,x)=f(x), \quad 0\leq x\leq L$
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$u(0,x)=f(x), \quad 0\leq x\leq L$
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lets choose $L=\pi$
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lets choose $L=\pi$
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$f(x)=\begin{cases}-x & 0\leq x\leq \frac{\pi}{2} \\1-x & \frac{\pi}{2}<x\leq \pi\end{cases}$
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$f(x)=\begin{cases}-x & 0\leq x\leq \frac{\pi}{2} \\1-x & \frac{\pi}{2}<x\leq \pi\end{cases}$
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![[Lec 30 2023-11-24 13.42.29.excalidraw]]
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So we have a non-uniformly heated rod with both ends insulated. What happens to the temperature inside the rod over time? <i>"\[...\]. Very interesting problem."</i> -Prof (I agree.)
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So we have a non-uniformly heated rod with both ends insulated. What happens to the temperature inside the rod over time? <i>"\[...\]. Very interesting problem."</i> -Prof (I agree.)
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If we made this a series, where would it converge? Well it's continuous from 0 to pi and its windowed form when repeated will be convergent everywhere, this is good news for us.
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If we made this a series, where would it converge? Well it's continuous from 0 to pi and its windowed form when repeated will be convergent everywhere, this is good news for us.
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Separation of variables: $u(t,x)=T(t)X(x)$
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Separation of variables: $u(t,x)=T(t)X(x)$
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@ -229,7 +229,7 @@ ok i spent 5 minutes talking nonsense but
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you complaining that your exams are hard theyre not hard. I'm talking about 40 years ago, no computers no cellphones, and it wasnt so bad because we had to use our brain more.
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you complaining that your exams are hard theyre not hard. I'm talking about 40 years ago, no computers no cellphones, and it wasnt so bad because we had to use our brain more.
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now we consider a guitar string:
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now we consider a guitar string:
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![[Partial differential equations (lec 30-32) 2023-12-01 13.49.58.excalidraw]]
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assuming the thickness of the string is much smaller than the length of the string, which is true.
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assuming the thickness of the string is much smaller than the length of the string, which is true.
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$\frac{ \partial u^2 }{ \partial t^2 }=\alpha^2 \frac{ \partial^2 u }{ \partial x^2 } \quad 0\leq x\leq L, t>0$
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$\frac{ \partial u^2 }{ \partial t^2 }=\alpha^2 \frac{ \partial^2 u }{ \partial x^2 } \quad 0\leq x\leq L, t>0$
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^ Reminds me of the wave equation from phys 130.
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^ Reminds me of the wave equation from phys 130.
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@ -189,7 +189,6 @@ if $p(x)$ and $q(x)$ are <u>analytic</u> functions in a vicinity of $x_{0}$ then
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we expect that the solution $y$ can be represented by a power series. This is true according to the following theorem:
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we expect that the solution $y$ can be represented by a power series. This is true according to the following theorem:
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Theorem: If $x_{0}$ is an ordinary point then the differential equation above has two linearly independent solution of the form $\sum_{n=0} ^\infty a_{n}(x-x_{0})^n, \qquad\sum_{n=0}^\infty b_{n}(x-x_{0})^n$.
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Theorem: If $x_{0}$ is an ordinary point then the differential equation above has two linearly independent solution of the form $\sum_{n=0} ^\infty a_{n}(x-x_{0})^n, \qquad\sum_{n=0}^\infty b_{n}(x-x_{0})^n$.
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The radius of convergence for them is at least as large as the distance between $x_{0}$ and the closest singular point (which can be real or complex).
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The radius of convergence for them is at least as large as the distance between $x_{0}$ and the closest singular point (which can be real or complex).
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## Examples for calculating $\rho$
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## Examples for calculating $\rho$
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#ex
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#ex
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@ -306,7 +305,7 @@ $\left( 30a_{6}+a_{3}-\cancelto{ 0 }{ \frac{a_{1}}{6} } \right)t^4=0 \implies a_
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</br>
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</br>
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substitute back $x-\pi=t$
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substitute back $x-\pi=t$
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$$y(x)=1-\frac{1}{6}(x-\pi)^3+\frac{1}{120}(x-\pi)^5+\frac{1}{180}(x-\pi)^6+\dots$$
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$$y(x)=1-\frac{1}{6}(x-\pi)^3+\frac{1}{120}(x-\pi)^5+\frac{1}{180}(x-\pi)^6+\dots$$
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there's no general formula here for the constants (or maybe he said no formula for y(x)?), but we can write the solution in the following form^.
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there's no general formula here for the constants (or maybe he said no formula for $y(x)$?), but we can write the solution in the following form^.
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#ex #powseries
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#ex #powseries
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Here's a non-homogenous example: (RHS$\ne 0$)
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Here's a non-homogenous example: (RHS$\ne 0$)
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@ -5,140 +5,185 @@ We start with some thermodynamics
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Heat equation not only describes thermodynamics but it can also model the diffusion of gasses. It is a partial differential equation.
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Heat equation not only describes thermodynamics but it can also model the diffusion of gasses. It is a partial differential equation.
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Strikingly, it can also model option prices in the stock market. However, using it as a strategy to make money is not so simple, because if it worked then everyone would try to use it to make money, which would cause the overall strategy to be less effective as the option prices start to get priced to accommodate for the prediction (🤯).
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Strikingly, it can also model option prices in the stock market. However, using it as a strategy to make money is not so simple, because if it worked then everyone would try to use it to make money, which would cause the overall strategy to be less effective as the option prices start to get priced to accommodate for the prediction (🤯).
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We assume that the tube is perfectly insulating along its surface, this helps reduce the problem into a one dimensional problem. Heat can only travel inside and along the x axis.
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We assume that the tube is perfectly insulating along its cylindrical surface, this helps reduce the problem into a one dimensional problem. Heat can only travel inside and along the x axis.
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We can express the heat along the tube as $u(t,x)$
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>I like to visualize $u(t,x)$ as a wavetable that smooths out as the time variable progresses. Something like this: [(image source)](https://markmoshermusic.com/2015/07/29/intro-to-loading-custom-waldorf-blofeld-wavetables/)
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## Derivations (I'd skip this)
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Fourier figured out that:
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Fourier figured out that:
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$\text{Heat flux} = -k(x)a\frac{\partial u}{\partial x}(t,x) \Delta t$
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$\text{Heat flux} = -k(x)a\frac{\partial u}{\partial x}(t,x) \Delta t$
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heat flux is in the positive $x$ direction
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heat flux is in the positive $x$ direction
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where du/dx is the opposite sign of the flux (because hot flows to cold.)
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$k(x)$ is the thermal conductivity of the tube at point $x$
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where $u(t,x)$ is a function that describes the temperature in the tube.
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$\frac{ \partial u }{ \partial x }$ is always opposite in sign of flux (because hot flows to cold.)
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$-k(x+\Delta x)a\frac{\partial u}{\partial x}(t,x+\Delta x) \Delta t$
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$a$ is the area of the cross-section.
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$=-k(x+\Delta x)a\frac{\partial u}{\partial x}(t,x+\Delta x) \Delta t$
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$\Delta u=u(t+\Delta t,x)-u(t,x)$
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$\Delta u=u(t+\Delta t,x)-u(t,x)$
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Amount of heat to change temperature over $\Delta t$ with $\Delta u$ is $\underbrace{ C(x) }_{ \text{specific heat capacity } }\underbrace{ \rho(x) }_{ \text{density} }\Delta u\underbrace{ a\Delta x }_{ \text{volume} }$
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Amount of heat to change temperature over $\Delta t$ with $\Delta u$ is $\underbrace{ C(x) }_{ \text{specific heat capacity } }\underbrace{ \rho(x) }_{ \text{density} }\Delta u\underbrace{ a\Delta x }_{ \text{volume} }$
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$c(x)\rho(x)a\Delta x(u(t+\Delta t,x)-u(t,x))=a\Delta t(k(x+\Delta x) \frac{\partial u}{\partial x}(t_{1}x+\Delta x)-k(x) \frac{\partial u}{\partial x}(t,x))+Q(t,x)\Delta xa\Delta t$
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$C(x)\rho(x)a\Delta x(u(t+\Delta t,x)-u(t,x))=a\Delta t(k(x+\Delta x) \frac{\partial u}{\partial x}(t,x+\Delta x)-k(x) \frac{\partial u}{\partial x}(t,x))+Q(t,x)\Delta xa\Delta t$
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>I'm gonna be honest, I'm lost in this derivation already. What is $Q(t,x)$?
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divide by $a\Delta x\Delta t$
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divide by $a\Delta x\Delta t$
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$c(x)\rho(x)\frac{(u(t+\Delta t,x)-u(t,x))}{\Delta t}=a\Delta t\left( k(x+\Delta x) \frac{\partial u}{\partial x}(t,x+\Delta x)-k(x) \frac{\partial u}{\partial x}(t,x) \right)+Q(t,x)$
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$C(x)\rho(x)\frac{(u(t+\Delta t,x)-u(t,x))}{\Delta t}=\frac{ k(x+\Delta x) \frac{\partial u}{\partial x}(t,x+\Delta x)-k(x) \frac{\partial u}{\partial x}(t,x)}{\nabla x}+Q(t,x)$
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$\lim_{ \Delta x,\Delta t \to 0 }: c(x)\rho(x)\frac{ \partial u }{ \partial t }(t,x)=\frac{ \partial }{ \partial x }\left( k(x)\frac{ \partial u }{ \partial x }(t,x) \right)+Q(t,x)$
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$\lim_{ \Delta x,\Delta t \to 0 }: C(x)\rho(x)\frac{ \partial u }{ \partial t }(t,x)=\frac{ \partial }{ \partial x }\left( k(x)\frac{ \partial u }{ \partial x }(t,x) \right)+Q(t,x)$
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Any differential equation, to his knowledge, is to balance some volume and to take the limit to produce a pointwise / instantaneous balance (in the form of a differential equation)
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Any differential equation, to his knowledge, is to balance some volume and to take the limit to produce a pointwise / instantaneous balance (in the form of a differential equation)
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the Maxwell equations that describes magnetic and electric fields are a system of partial differential equations.
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the Maxwell equations that describes magnetic and electric fields are a system of partial differential equations.
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thermodynamics can be very important for electrical engineers, for instance the heat produced in a transformer, or a battery. It has applications.
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thermodynamics can be very important for electrical engineers, for instance the heat produced in a transformer, or a battery. It has applications.
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## Separation of variables & Eigen value problems
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#ex #SoV #evp
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(This is more of a case study than an example.)
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Rewrite the equation we derived by grouping the constant terms into one constant $D$
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$\frac{ \partial u }{ \partial t }=D \frac{ \partial^2 u }{ \partial x^2 }, \quad 0\leq x\leq L, \quad t>0$
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$\frac{ \partial u }{ \partial t }=D \frac{ \partial^2 u }{ \partial x^2 }, \quad 0\leq x\leq L, \quad t>0$
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boundary conditions:
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boundary conditions:
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$u(t,0)=u(t,L)=0 , \quad t>0$ (simple case)
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$u(t,0)=u(t,L)=0 , \quad t>0$ (simple case)
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$u(0,x)=f(x) , \quad 0\leq x\leq L$
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$u(0,x)=f(x) , \quad 0\leq x\leq L$
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These three equations form an IBVP (initial boundary value problem)
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These three equations form an IBVP (initial boundary value problem)
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we will work on this equation next lecture:
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Separation of variables starts with the assumption that the solution can be written as:
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$u(t,x)=T(t)X(x)$
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$u(t,x)=T(t)X(x)$
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$T'x=DTx''$
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plug into equation:
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divide by DTx:
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$\frac{ \partial}{ \partial t }(T(t)X(x))=D \frac{ \partial^2 }{ \partial x^2 }(T(t)X(x))$
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$\frac{T'}{DT}=\frac{x''}{x}$
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$T'X=DTX''$
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on the left is a function of time only, on the right is a function of x only.
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divide by $DTX$:
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$\frac{T'}{DT}=\frac{x''}{x}=-\lambda$ where $\lambda$ is a constant
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$\frac{T'}{DT}=\frac{X''}{X}$
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$x''+\lambda x=0$
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recall $T$ and $X$ are single variable functions ($u(t,x)=T(t)X(x)$)
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$u(t,0)=\cancel{ T(t) } \quad x(0)=0$
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that means the left is a function of $t$ only, on the right is a function of $x$ only.
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$x(0)=x(L)=0$
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$\frac{T'}{DT}=\frac{X''}{X}=-\lambda$ where $\lambda$ is a constant
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this is called an eigen value problem.
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we first consider the problem:
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$\frac{X''}{X}=-\lambda$
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$X''+\lambda X=0$ (called an eigen value problem on a second derivative operator)
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$u(t,0)=0=T(t)X(0)$
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$\implies X(0)=0$
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$T(t)$ cannot be the 0 term because if it was then you get $u(t,x)=T(t)X(x)=0$ for all $t$ and $x$ which can't be.
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$u(t,L)=0=T(t)X(L)$
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$\implies X(L)=0$
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</br>
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$\begin{cases}X''+\lambda X=0 \\X(0)=0 \\X(L)=0\end{cases}$ <-This is called an eigen value problem.
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#end of lec 26 #start of lec 27
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#end of lec 26 #start of lec 27
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Recall from last lec:
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$\frac{ \partial u }{ \partial t }=D \frac{ \partial^2 u }{ \partial x^2 }, \quad 0\leq x\leq L, \quad t>0$
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It's an eigen value problem because unlike before when we did second order linear equations with constant coefficients, lambda is not fixed, it can be any number.
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boundary conditions:
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recall in linear algebra $Ax+\lambda x=0$ where $A$ is a matrix
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$u(t,0)=u(t,L)=0 , \quad t>0$ (simple case)
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eigen is a Germanic word meaning intrinsic, important
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$u(0,x)=f(x) , \quad 0\leq x\leq L$
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We find the characteristic equations by splitting into three cases, just like we've done since lecture 7
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This is an IBVP
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$X''+\lambda X=0$
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$\frac{T'}{DT}=\frac{x''}{x}=-\lambda$
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case 1) $\lambda<0$
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$x''+\lambda x=0$ (called an eigen value problem on a second derivative operator)
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$r^2+\lambda=0 \implies r_{1,2}=\pm \sqrt{ -\lambda }$ (real solutions only since $\sqrt{ -(-) }=\sqrt{ + }$)
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$x(0)=x(L)=0$
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$X(x)=c_{1}e^{\sqrt{ -\lambda }x}+c_{2}e^{-\sqrt{ -\lambda }x}$
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its an eigen value problem because lambda is not fixed, it can be any number.
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recall in linear algebra $Ax+\lambda x=0$ where A is a matrix
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eigen is a germanic word meaning intrinsic, important
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(i)$\lambda<0$
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$r^2+\lambda=0 \implies r_{1,2}=\pm \sqrt{ -\lambda }$
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$X(x)=c_{1}e^{\sqrt{ -\lambda }x}+c_{2}^{-\sqrt{ -\lambda }x}$
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but we have a boundary condition:
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but we have a boundary condition:
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$X(0)=0=c_{1}+c_{2}$
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$c_{1}+c_{2}=0$
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$c_{1}+c_{2}=0$
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$X(L)=c_{1}e^{\sqrt{ -\lambda }L}+c_{2}^{-\sqrt{ -\lambda }L}=0$
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$X(L)=c_{1}e^{\sqrt{ -\lambda }L}+c_{2}e^{-\sqrt{ -\lambda }L}=0$
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this has a solution, as the determinant $\ne 0$
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this has a unique solution, as the determinant is non-zero: $\det\left(\begin{matrix}1 & 1 \\e^{\sqrt{ -\lambda }L} & e^{-\sqrt{ -\lambda }L}\end{matrix}\right)=e^{-\sqrt{ -\lambda }L}-e^{\sqrt{ -\lambda }L}\ne 0$ (as long as $L\ne 0$, which is true since we are assuming the tube has non-zero length.)
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the solution is $c_{1}=0,\ c_{2}=0$
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the only solution is $c_{1}=0,\ c_{2}=0$
|
||||||
which gives $X(x)=0$ for all $x\in[0,L]$ very boring solution!
|
which gives $X(x)=0$ for all $x\in[0,L]$ very boring solution!
|
||||||
other option:
|
</br>
|
||||||
(ii) $\lambda=0$
|
case 2) $\lambda=0$
|
||||||
$x''=0$
|
$X''=0$
|
||||||
integrate both sides twice:
|
$r^2=0$
|
||||||
$X(x)=c_{1}x+c_{2}$
|
$r=0$ (repeated root)
|
||||||
boundary conditions:
|
$X(x)=c_{1}e^{0}+c_{2}0e^0=c_{1}$
|
||||||
$X(0)=0\implies c_{2}=0$
|
$X(0)=0\implies c_{1}=0$
|
||||||
$X(L)=0=c_{1}L \implies c_{1}=0$
|
$X(L)=0 \implies c_{1}=0$
|
||||||
again a boring solution! $X(x)=0$ this is called a trivial solution.
|
there are no restrictions on $c_{2}$
|
||||||
We are looking for non-trival solutions.
|
but our solution is $X(x)=0$ again for all $x$. This is called a trivial solution.
|
||||||
try the last other possible case:
|
>alternatively, we could have found this in a different approach:
|
||||||
(iii) $\lambda>0$
|
>$X''=0$
|
||||||
$r_{1,2}=\pm \sqrt{ \lambda }i$
|
>integrate both sides twice:
|
||||||
|
>$X(x)=c_{1}x+c_{2}$
|
||||||
|
>boundary conditions:
|
||||||
|
>$X(0)=0\implies c_{2}=0$
|
||||||
|
>$X(L)=0=c_{1}L \implies c_{1}=0$
|
||||||
|
>$\implies X(x)=0$
|
||||||
|
>This method worked because the LHS is just one term of $X^n$ so we treated it as a separable equation.
|
||||||
|
|
||||||
|
We are looking for non-trival solutions. Try the last other possible case:
|
||||||
|
case 3) $\lambda>0$
|
||||||
|
$r^2+\lambda=0$
|
||||||
|
$r_{1,2}=\pm \sqrt{ -\lambda }=\pm i\sqrt{ \lambda }$
|
||||||
$X(x)=c_{1}\cos(\sqrt{ \lambda }x)+c_{2}\sin(\sqrt{ \lambda }x)$
|
$X(x)=c_{1}\cos(\sqrt{ \lambda }x)+c_{2}\sin(\sqrt{ \lambda }x)$
|
||||||
boundary conditions:
|
boundary conditions:
|
||||||
$X(0)=0\implies c_{1}=0$
|
$X(0)=0\implies c_{1}=0$
|
||||||
$X(L)=0 \implies c_{2}\sin(\sqrt{ \lambda }L)=0$
|
$X(L)=0 \implies c_{2}\sin(\sqrt{ \lambda }L)=0$
|
||||||
if you take $c_{2}=0$ you lose, another boring solution.
|
if you take $c_{2}=0$ you lose, another boring solution ($X(x)=0$).
|
||||||
but if $\sin(\sqrt{ \lambda }L)=0 \implies \sqrt{ \lambda }L=n\pi$
|
but if $\sin(\sqrt{ \lambda }L)=0 \implies \sqrt{ \lambda }L=n\pi$
|
||||||
where $n=1,2,3,\dots$ notice that $n\ne 0$ becuase that implies lambda=0 but lambda is >0 so that cant be.
|
where $n=1,2,3,\dots$ notice that $n\ne 0$ because that implies $\lambda=0$ but $\lambda>0$ in case 3 so that cant be.
|
||||||
$\lambda_{n}=\frac{n\pi}{L}$
|
$\lambda_{n}=(\frac{n\pi}{L})^2$ <- These are called <u>eigen values</u>.
|
||||||
$X_{n}(x)=c_{2}\sin(\sqrt{ \lambda }x)=\sin\left( \frac{n\pi}{L}x \right)$ this is an eigen function. It's the only non-trivial solution.
|
The corresponding <u>eigen functions</u> are:
|
||||||
|
$X_{n}(x)=c_{2}\sin(\sqrt{ \lambda_{n} }x)=c_{n}\sin\left( \frac{n\pi}{L}x \right)$
|
||||||
lets go back to the problem:
|
^They are the only non-trivial solutions.
|
||||||
$\frac{T'}{DT}=\frac{x''}{x}=-\lambda$
|
</br>
|
||||||
|
lets go back to the problem and focus on $T$:
|
||||||
|
$\frac{T'}{DT}=\frac{X''}{X}=-\lambda$
|
||||||
$\frac{T'}{T}=-\left( \frac{n\pi}{L} \right)^2D$
|
$\frac{T'}{T}=-\left( \frac{n\pi}{L} \right)^2D$
|
||||||
this is a separable equation.
|
this is a separable equation.
|
||||||
$T_{n}(t)=c_{n}e^{-(n\pi/2)^2Dt}$
|
We can treat the function T as a variable:
|
||||||
you might have thought you could forget the material before the midterm to make more space in your brain, your brains are a lot more emptier than mine, mine is filled with garbage [...] so don't forget anything you learned before the midterm!
|
$\frac{dT}{dt} \frac{1}{T}=-\left( \frac{n\pi}{L} \right)^2D$
|
||||||
you'll see that as you continue in education, you'll see a lot of completely new things and you'll have to find shortcuts and tricks to make the content fit what you already know.
|
$\int{dT} \frac{1}{T}=\int-\left( \frac{n\pi}{L} \right)^2Ddt$
|
||||||
okay enough with life stuff, back to mathematics:
|
$\ln(T)=-\left( \frac{n\pi}{L} \right)Dt+c_{n}$
|
||||||
$\implies u_{n}(t,x)=c_{n}e^{-(n\pi/2)^2Dt}\sin\left( \frac{n\pi x}{L} \right)$ $n=1,2,3,\dots$
|
$T_{n}(t)=c_{n}e^{-(\frac{n\pi}{L})^2Dt}$
|
||||||
applying superposition (sum of any solutions is also a solution)
|
>Yes this looks illegal, but it works, you could also integrate more rigorously if you did a u-sub: $u=T(t) \quad \frac{du}{dt}=T'(t)$)
|
||||||
$u(t,x)=\sum_{n=1}^\infty c_{n}e^{-(n\pi/L)^2Dt}\sin\left( \frac{n\pi x}{L} \right)$ this is the most general form of the solution.
|
</br>
|
||||||
$u(0,x)=f(x)=\sum_{n=1}^\infty c_{n}\sin\left( \frac{n\pi x}{L} \right),\ 0\leq x\leq L$ we have never seen anything like this before, an infinite number of sin terms added together, usually its polynonmials we sum.
|
><i>"You might have thought you could forget the material before the midterm to make more space in your brain, your brains are a lot more emptier than mine, (class laughs) mine is filled with garbage [...] so don't forget anything you learned before the midterm!"</i>
|
||||||
this is when fourier (?) steps in. he proved that you can represent various functions as a sum of sines.
|
><i>"You'll see that as you continue in education, you'll see a lot of completely new things and you'll have to find shortcuts and tricks to make the content fit what you already know."</i>
|
||||||
fun stories about pioncre and cauchy euler: cauchy euler was an engineer, and pioncre had his theorem released around 2008 and it was really long, like 400 pages, he posted it online and asked if anyone wanted to prove it, after a while 4-5 or so mathematicians checked his proof and said, yep okay the proof looks correct. His theorem has a lot to do with the material world.
|
|
||||||
f doesn't even have to be analytic in order for it to be expressed as a sum of sines.
|
|
||||||
|
|
||||||
take $f(x)=\sin(x)$ and $L=\pi$ can we pick the coefficients $c_{n}$ to make the expressian above true?
|
Okay enough with life stuff, back to mathematics:
|
||||||
of course $c_{1}=1$ and all the other coefficients equals 0
|
$\implies u_{n}(t,x)=c_{n}e^{-(n\pi/L)^2Dt}\sin\left( \frac{n\pi x}{L} \right)$ $n=1,2,3,\dots$
|
||||||
|
applying superposition (sum of any solutions is also a solution):
|
||||||
done with class! more history time:
|
$$u(t,x)=\sum_{n=1}^\infty c_{n}e^{-(n\pi/L)^2Dt}\sin\left( \frac{n\pi x}{L} \right)$$ ^This is the most general form of the solution.
|
||||||
why is this important? in 1979 a team of engineers and mathematicians from a company philips they discovered, or practically implemented: that an audio signal has billions of datapoints over time if you represent it as a fourier seriers, and truncate some of the coeffecients we can represent many signals really well. which condenses down the data to just a handful of coefficients
|
$u(0,x)=f(x)=\sum_{n=1}^\infty c_{n}\sin\left( \frac{n\pi x}{L} \right),\ 0\leq x\leq L$ we have never seen anything like this before, an infinite number of sin terms added together, usually its polynomials that are summed.
|
||||||
this is how EQ's are made too to filter out noise, just set the c_n of the frequencies you dont want to 0
|
This is when Fourier (?) steps in. He proved that you can represent various functions as a sum of sines.
|
||||||
so philips used math, math that is simillar to what we are discussing in the lecture, to make a digital record, the first digital cd. Not only that but fourier series are used for image and video compression as well, although they often use a sum of wavelets instead of a sum of trigonometric functions.
|
fun stories about Pioncre and Cauchy Euler: Cauchy Euler was an engineer, and Pioncre had his theorem released around 2008 and it was really long, like 400 pages, he posted it online and asked if anyone wanted to prove it, after a while 4-5 or so mathematicians checked his proof and said, yep okay the proof looks correct. His theorem has a lot to do with the material world.
|
||||||
#end of lec 27
|
</br>
|
||||||
<i>reading week</i>
|
$f$ doesn't even have to be analytic in order for it to be expressed as a sum of sines.
|
||||||
|
Take $f(x)=\sin(x)$ and $L=\pi$ can we pick the coefficients $c_{n}$ to make the expression above true?
|
||||||
|
of course! $c_{1}=1$ and all the other coefficients equals 0.
|
||||||
|
</br>
|
||||||
|
Done with class! More history time:
|
||||||
|
Why is this important? in 1979 a team of engineers and mathematicians from a company Philips they discovered, or practically implemented that an audio signal has billions of datapoints over time if you represent it as a Fourier series, and truncate some of the coefficients we can represent many signals really well. Which condenses down the data to just a handful of coefficients
|
||||||
|
This is how filters are made too to filter out noise, just set the $c_{n}$ of the frequencies you don't want to $0$.
|
||||||
|
So Philips used math, math that is similar to what we are discussing in the lecture, to make a digital record, the first digital cd. Not only that but Fourier series are used for image and video compression as well, although they often use a sum of wavelets instead of a sum of trigonometric functions.
|
||||||
|
#end of lec 27 (Nov 10)
|
||||||
|
<i>*reading week*</i>
|
||||||
#start of lec 28 (Nov 20)
|
#start of lec 28 (Nov 20)
|
||||||
|
## Example Eigen value problem
|
||||||
|
#ex #evp
|
||||||
$y''-2y'+\lambda y=0 \qquad y(0)=0, \quad y'(\pi)+y(\pi)=0$
|
$y''-2y'+\lambda y=0 \qquad y(0)=0, \quad y'(\pi)+y(\pi)=0$
|
||||||
eigen value problem, find $\lambda$ such that the initial conditions are true.
|
Find $\lambda$ such that the initial conditions are true.
|
||||||
find the characteristic polynomial.
|
find the characteristic polynomial:
|
||||||
$r^2-2r+\lambda=0$
|
$r^2-2r+\lambda=0$
|
||||||
|
|
||||||
$r_{1,2}=\frac{2\pm \sqrt{ 4-4\lambda }}{2}=1\pm \sqrt{ 1-\lambda }$
|
$r_{1,2}=\frac{2\pm \sqrt{ 4-4\lambda }}{2}=1\pm \sqrt{ 1-\lambda }$
|
||||||
|
</br>
|
||||||
case 1: $1-\lambda>0$
|
case 1: $1-\lambda>0$
|
||||||
$y(x)=c_{1}e^{(1+\sqrt{ 1-\lambda })x}+c_{2}e^{ (1-\sqrt{ 1-\lambda })x }$
|
$y(x)=c_{1}e^{(1+\sqrt{ 1-\lambda })x}+c_{2}e^{ (1-\sqrt{ 1-\lambda })x }$
|
||||||
$y(0)=0=c_{1}+c_{2}$
|
$y(0)=0=c_{1}+c_{2}$
|
||||||
$c_{1}(1+\sqrt{ 1-\lambda })e^{(1+\sqrt{ 1-\lambda })\pi}+c_{2}(1-\sqrt{ 1-\lambda })e^{ (1-\sqrt{ 1-\lambda })\pi }+c_{1}e^{(1+\sqrt{ 1-\lambda })\pi}+c_{2}e^{ (1-\sqrt{ 1-\lambda })\pi }=0$
|
$y'(\pi)+y(\pi)=0$
|
||||||
|
$\implies c_{1}(1+\sqrt{ 1-\lambda })e^{(1+\sqrt{ 1-\lambda })\pi}+c_{2}(1-\sqrt{ 1-\lambda })e^{ (1-\sqrt{ 1-\lambda })\pi }+c_{1}e^{(1+\sqrt{ 1-\lambda })\pi}+c_{2}e^{ (1-\sqrt{ 1-\lambda })\pi }=0$
|
||||||
|
substitute $c_{1}=-c_{2}$
|
||||||
$-c_{2}(1+\sqrt{ 1-\lambda })e^{(1+\sqrt{ 1-\lambda })\pi}+c_{2}(1-\sqrt{ 1-\lambda })e^{ (1-\sqrt{ 1-\lambda })\pi }-c_{2}e^{(1+\sqrt{ 1-\lambda })\pi}+c_{2}e^{ (1-\sqrt{ 1-\lambda })\pi }=0$
|
$-c_{2}(1+\sqrt{ 1-\lambda })e^{(1+\sqrt{ 1-\lambda })\pi}+c_{2}(1-\sqrt{ 1-\lambda })e^{ (1-\sqrt{ 1-\lambda })\pi }-c_{2}e^{(1+\sqrt{ 1-\lambda })\pi}+c_{2}e^{ (1-\sqrt{ 1-\lambda })\pi }=0$
|
||||||
$c_{2}=0=c_{1}$ boring solution :( (also called a trivial solution)
|
$\implies c_{2}=0$
|
||||||
|
$\implies y(x)=0$ boring solution :( (also called a trivial solution)
|
||||||
|
</br>
|
||||||
case 2: $1-\lambda=0$
|
case 2: $1-\lambda=0$
|
||||||
$y(x)=c_{1}e^x+c_{2}xe^x$
|
$r=r_{1,2}=1\pm \sqrt{ 0 }$ (repeated root)
|
||||||
|
$y(x)=c_{1}e^{rx}+c_{2}xe^{rx}$
|
||||||
$y(0)=0=c_{1}$
|
$y(0)=0=c_{1}$
|
||||||
$y'(\pi)+y(\pi)=c_{2}(\pi e^\pi+e^\pi+\pi e^\pi)=0 \implies c_{2}=0$
|
$y'(\pi)+y(\pi)=c_{2}(\pi e^\pi+e^\pi+\pi e^\pi)=0 \implies c_{2}=0$
|
||||||
(same trivial solution.)
|
(same trivial solution.)
|
||||||
|
</br>
|
||||||
case 3: $1-\lambda<0$
|
case 3: $1-\lambda<0$
|
||||||
$r_{1,2}=1\pm i\sqrt{\lambda-1}$
|
$r_{1,2}=1\pm i\sqrt{\lambda-1}$
|
||||||
|
</br>
|
||||||
$y(x)=e^x(c_{1}\cos(\sqrt{ \lambda-1 }x)+c_{2}\sin(\sqrt{ \lambda-1 }x))$
|
$y(x)=e^x(c_{1}\cos(\sqrt{ \lambda-1 }x)+c_{2}\sin(\sqrt{ \lambda-1 }x))$
|
||||||
$y(0)=0=c_{1}$
|
$y(0)=0=c_{1}$
|
||||||
$y(x)=c_{2}e^x\sin \sqrt{ \lambda-1 }x$
|
$y(x)=c_{2}e^x\sin (\sqrt{ \lambda-1 }x)$
|
||||||
$y'(\pi)+y(\pi)=0=c_{2}(e^\pi \sin \sqrt{ \lambda-1 }\pi+e^\pi \cos( \sqrt{ x-1 }\pi)\sqrt{ \lambda-1 })+c_{2}e^\pi \sin \sqrt{ \lambda-1 }\pi$
|
$y'(\pi)+y(\pi)=0=c_{2}\underbrace{ (e^\pi \sin \sqrt{ \lambda-1 }\pi+e^\pi \cos( \sqrt{ x-1 }\pi)\sqrt{ \lambda-1 }+e^\pi \sin \sqrt{ \lambda-1 }\pi) }_{ =0 }$
|
||||||
$$2\sin(\sqrt{ \lambda-1 }\pi)-\sqrt{ \lambda-1 }\cos(\sqrt{ \lambda-1 }\pi)=0$$
|
$$2\sin(\sqrt{ \lambda-1 }\pi)+\sqrt{ \lambda-1 }\cos(\sqrt{ \lambda-1 }\pi)=\frac{0}{e^\pi}=0$$
|
||||||
This is a transcendental equation, non algebraic, cannot be solved explicitly.
|
This is a transcendental equation, non algebraic, cannot be solved explicitly.
|
||||||
Only option we have is to approximate the eigen values. there are an infinite number of them.
|
Btw the eigen function is $y(x)=ce^x\sin(\sqrt{\lambda-1}x)$
|
||||||
We have to use software in order to obtain a finite number of approximations. I like software, software is good, as long as your being mindful of how you're using it.
|
Only option we have is to approximate the eigen values. There are an infinite number of them.
|
||||||
"engineers over design things." first they use software and add a 50%, sometimes 300% margin on it. But if your using that big of a margin I can just tell you how far you need to build your house away from the river. -Prof
|
We have to use software in order to obtain a finite number of approximations. I like software, software is good, as long as your being mindful of how you're using it. (I'm not putting quotes as I can't recall for certain what he said.)
|
||||||
|
"Engineers over design things." first they use software and add a 50%, sometimes 300% margin on it. But if your using that big of a margin I can just tell you how far you need to build your house away from the river. -Prof
|
||||||
|
</br>
|
||||||
|
Here's a plot, every crossing of red line with the $x$ axis is a solution for $\lambda$. Blue is a solution curve corresponding with $\lambda\approx8.305027$. Green shows the DE is being satisfied. And the equation on the bottom left shows the second initial condition is being met. The first condition is also met as you can see $Y(0)=0$ on the plot.
|
||||||
|
|
||||||
We are done.
|
We are done.
|
||||||
|
|
@ -70,7 +70,7 @@ $y'=z$
|
||||||
$\begin{cases}y'-z=0 \\az'+bz+cy=f\end{cases}$
|
$\begin{cases}y'-z=0 \\az'+bz+cy=f\end{cases}$
|
||||||
|
|
||||||
last example of the chapter:
|
last example of the chapter:
|
||||||
#ex
|
#ex #SoLE #IVP
|
||||||
$x'+y=0, \qquad x(0)=0$
|
$x'+y=0, \qquad x(0)=0$
|
||||||
$x+y'=1-u(t-2) \qquad y(0)=0$
|
$x+y'=1-u(t-2) \qquad y(0)=0$
|
||||||
This is a review problem in chapter 7 of the textbook.
|
This is a review problem in chapter 7 of the textbook.
|
||||||
|
|
|
||||||
|
|
@ -23,7 +23,7 @@ I have written these notes for myself, I thought it would be cool to share them.
|
||||||
[Dirak δ-function (lec 21)](dirak-δ-function-lec-21.html)
|
[Dirak δ-function (lec 21)](dirak-δ-function-lec-21.html)
|
||||||
[Systems of linear equations (lec 21-22)](systems-of-linear-equations-lec-21-22.html)
|
[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)
|
[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) (raw notes, not reviewed or revised yet.)
|
[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.)
|
[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.)
|
[Partial differential equations (lec 30-33)](partial-differential-equations-lec-30-33.html) (raw notes, not reviewed or revised yet.)
|
||||||
</br>
|
</br>
|
||||||
|
|
|
||||||
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@ -0,0 +1,762 @@
|
||||||
|
---
|
||||||
|
|
||||||
|
excalidraw-plugin: parsed
|
||||||
|
tags: [excalidraw]
|
||||||
|
|
||||||
|
---
|
||||||
|
==⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠==
|
||||||
|
|
||||||
|
|
||||||
|
# Text Elements
|
||||||
|
%%
|
||||||
|
# Drawing
|
||||||
|
```compressed-json
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|
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||||||
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||||||
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||||||
|
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||||||
|
|
||||||
|
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||||||
|
|
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|
||||||
|
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||||||
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|
||||||
|
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||||||
|
|
||||||
|
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||||||
|
|
||||||
|
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|
||||||
|
|
||||||
|
dVeIxabpSTGTJVz4+oUnFpnFIoGipwU44diXrA5SXLAL/ZkOi9l7cZftjDXmWPGU
|
||||||
|
|
||||||
|
mSFe3DjBuZl4f5l3BuuLv7t2TwbOX0I3diYVRyMEPZUqQdNdzUuVpNR5X81rNUkT
|
||||||
|
|
||||||
|
Ah4ABdBL8cAHADGgjMNwB6w0ADiBZAlQEQDE46wAwCEACABQDDAyZUwECRAoAsAJ
|
||||||
|
|
||||||
|
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|
||||||
|
|
||||||
|
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||||||
|
|
||||||
|
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|
||||||
|
|
||||||
|
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|
||||||
|
|
||||||
|
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|
||||||
|
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|
||||||
|
|
||||||
|
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||||||
|
|
||||||
|
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||||||
|
|
||||||
|
7+oALyNCoEwCQQygMI1lA2QLgCaAwQOwjVVEINgBEASJmgDeNdYBwCACfZaQADl4
|
||||||
|
|
||||||
|
YEIBwhkEB7DVV2sOAB2SeuAaDp6wAFrAgAWsEAA=
|
||||||
|
```
|
||||||
|
%%
|
||||||
Binary file not shown.
|
After Width: | Height: | Size: 8.2 KiB |
|
|
@ -0,0 +1,796 @@
|
||||||
|
---
|
||||||
|
|
||||||
|
excalidraw-plugin: parsed
|
||||||
|
tags: [excalidraw]
|
||||||
|
|
||||||
|
---
|
||||||
|
==⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠==
|
||||||
|
|
||||||
|
|
||||||
|
# Text Elements
|
||||||
|
%%
|
||||||
|
# Drawing
|
||||||
|
```compressed-json
|
||||||
|
N4KAkARALgngDgUwgLgAQQQDwMYEMA2AlgCYBOuA7hADTgQBuCpAzoQPYB2KqATL
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||||||
|
|
||||||
|
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||||||
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||||||
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|
||||||
|
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||||||
|
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||||||
|
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||||||
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||||||
|
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||||||
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|
||||||
|
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||||||
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|
||||||
|
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
|
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||||||
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|
||||||
|
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||||||
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|
||||||
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||||||
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||||||
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||||||
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||||||
|
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||||||
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||||||
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||||||
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||||||
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|
||||||
|
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||||||
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||||||
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||||||
|
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||||||
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|
||||||
|
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||||||
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||||||
|
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||||||
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||||||
|
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||||||
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|
||||||
|
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||||||
|
|
||||||
|
dnIEucCHgKGf7EZcSET8y6TTA4slZk3z3k44ZoTYRNlWcidZWctnuTzmA9BnkqRn
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||||||
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|
||||||
|
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||||||
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|
||||||
|
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||||||
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||||||
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|
||||||
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||||||
|
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||||||
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||||||
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||||||
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keoxo4gZoptDMeCec34nsg8sGVchsjc08/tIdEdMMMYiAZQNkHgUYHdKYGASQTQV
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|
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|
6h/Ew0oSAnrWaWA6XQ/fC9Pb3HpeKPpAss7e6gcpMmWZIv6aRFU+ROMk6ucic5Mi
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|
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||||||
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|
||||||
|
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||||||
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||||||
|
Wz2isz6cyVWbGlKbWQqK/QEavVagOZS94ckNSv/VhPaemMsuGm2Qu1Svo9gsuliB
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|
|
||||||
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|
||||||
|
|
||||||
|
2SIz6KxSI2ZXCz8zuDOgI2GEA3VMleaE1McrrecIGqQpbPaB8l1D6S+4+kRBQsIg
|
||||||
|
|
||||||
|
6Z8De6+8iaHMQruVQ8C60SC3Ncop5WC0CWahCuogwBomQJokMm81EWVWu1esGQfW
|
||||||
|
|
||||||
|
OCu2/G5QY0i8tcipoOAVkZwKAIYA0ZcA0NoAAQX0GcB6AACUEghBYg2g1190wRD0
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||||||
|
|
||||||
|
lgVg+KwQziRoyEYZ3YvEHoxLr0vMCpn7usSJwNUQPj6SwSa7i666gToNQSzNJ5sl
|
||||||
|
|
||||||
|
1z8iGajKTL6QfLHKUT8N0SOriMsSyNFRcSqMaNCS3KmMyTKgKSqSrRArfKviYqAr
|
||||||
|
|
||||||
|
aTgrhNfRwqJNcq+MwxeT4roqmry0hSVNRT1M8x0rTMdNwrywoqIn8qQJtsUh4Q0Q
|
||||||
|
|
||||||
|
yrNTeBksbMaq3NoryQGqVp/Gx0WqEBzSgHLSgtbTQsTxQzUAYmot+qin3Tnw8Uxo
|
||||||
|
|
||||||
|
xJvxxqqmMty0stprAySnIIQyhClrSt6swp1rSRNrw6u8RnetYbFYVq7bUd79T9ta
|
||||||
|
|
||||||
|
zr5UvqHqSdvtdt4z8LjyPUq7ecU8xosaGbSgo5Gwfhb6WzwCdd7U0jTIWoJIh7cZ
|
||||||
|
|
||||||
|
cLLcXobnsDkYyRSJHwx7LdL7yR7yxzfIpHzg66Os3rRnjr2CAD35Yp/o1swWuUIX
|
||||||
|
|
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|
||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
|
|
||||||
|
EzqdpAGuXSYQgFAA/gR6EbVDo4AKOwNlwQOmDAAg6CACDoQAA===
|
||||||
|
```
|
||||||
|
%%
|
||||||
Binary file not shown.
|
After Width: | Height: | Size: 25 KiB |
|
|
@ -0,0 +1,788 @@
|
||||||
|
---
|
||||||
|
|
||||||
|
excalidraw-plugin: parsed
|
||||||
|
tags: [excalidraw]
|
||||||
|
|
||||||
|
---
|
||||||
|
==⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠==
|
||||||
|
|
||||||
|
|
||||||
|
# Text Elements
|
||||||
|
%%
|
||||||
|
# Drawing
|
||||||
|
```compressed-json
|
||||||
|
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||||||
|
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||||||
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||||||
|
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||||||
|
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||||||
|
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||||||
|
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||||||
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||||||
|
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||||||
|
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||||||
|
|
||||||
|
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
|
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||||||
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||||||
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||||||
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|
||||||
|
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||||||
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||||||
|
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||||||
|
|
||||||
|
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||||||
|
|
||||||
|
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||||||
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||||||
|
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||||||
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||||||
|
qREZddRVdkJYJxslAsZF3Zi2fgI+RwirSmIfsA2eFjc7xQ/sPH6hjBHGJcWI8xQI
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||||||
|
|
||||||
|
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||||||
|
|
||||||
|
kpxhrTXtXRxKiTHBMfq9D6fVGw2SUmjSysSuYJIbEk9hZDSg5gwe+WG0MmaX2it0
|
||||||
|
|
||||||
|
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||||||
|
|
||||||
|
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||||||
|
|
||||||
|
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|
||||||
|
|
||||||
|
gwuyolXGOSikLPRQRPCTYMw6m2vMVU+D4GJWSkbW5CTjneWfBNRlQlf78X/hXZB3
|
||||||
|
|
||||||
|
LKUYomktSG8VOiAMVRbLJd9qloo1V5GURFnm5Rqkys2hrb5VLtlKgsMofL9V3nYg
|
||||||
|
|
||||||
|
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||||||
|
|
||||||
|
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||||||
|
|
||||||
|
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||||||
|
|
||||||
|
pg3jQ4ZMV62GMYSSlh5KY01hPdmdOzYd0MOJcwslbCo7dGehAfAoQoBOn0PoNQd5
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||||||
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|
||||||
|
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|
||||||
|
|
||||||
|
kzSNNnla89SYxuRN0UepdZWVpS0aBHheUiplTN3wxbdmg1i7MZmOStinG2MwLExb
|
||||||
|
|
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c21UW4egDoAkD5t+bUZxGcfoMgIIAygPBgxah4FDQ1txbRAARt8LVG2Yt6IOS1QA
|
||||||
|
|
||||||
|
rQIWWqtLwmYDCAzAOiHJtnLZUAIShlQgBltoYEwCwQygH60Dg2QBdnBAOeCiD7Ze
|
||||||
|
|
||||||
|
wNgBEAo6WgALtddhwAxi3AOu2OSQgFADfglQOu3vY4AMxxRuoIIk7AAb2CABvYQA
|
||||||
|
|
||||||
|
A===
|
||||||
|
```
|
||||||
|
%%
|
||||||
Binary file not shown.
|
After Width: | Height: | Size: 23 KiB |
|
|
@ -0,0 +1,778 @@
|
||||||
|
---
|
||||||
|
|
||||||
|
excalidraw-plugin: parsed
|
||||||
|
tags: [excalidraw]
|
||||||
|
|
||||||
|
---
|
||||||
|
==⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠==
|
||||||
|
|
||||||
|
|
||||||
|
# Text Elements
|
||||||
|
%%
|
||||||
|
# Drawing
|
||||||
|
```compressed-json
|
||||||
|
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|
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||||||
|
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||||||
|
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||||||
|
|
||||||
|
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||||||
|
|
||||||
|
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|
||||||
|
|
||||||
|
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|
||||||
|
|
||||||
|
CAAwmx8GxSJUAMSxBDc7lpUqQTS4bB05S0oQcYjM1nsiQ06zMOC4QI5fllZqEfD4
|
||||||
|
|
||||||
|
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||||||
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|
||||||
|
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
|
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||||||
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||||||
|
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||||||
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||||||
|
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
|
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||||||
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||||||
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||||||
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||||||
|
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||||||
|
|
||||||
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||||||
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||||||
|
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||||||
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|
||||||
|
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||||||
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||||||
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||||||
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|
||||||
|
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||||||
|
|
||||||
|
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||||||
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||||||
|
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||||||
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||||||
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||||||
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||||||
|
niJqnJ5JprqqjoD4upXLTjJrVq2k7JrhwRE1TxEI1qOzzcAzHjiK54RIoYBSBrZT
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|
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||||||
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||||||
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jtEwGJ4J/h2J2zydsFW98dTwsbPa6wgst94INJzYuVyd1NBsccNIHqEcAqEdOIVp
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||||||
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||||||
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||||||
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i8F656a9l6w756g8J7u8ZTx8e8t7is97l81kI0g7jp+6i8soSbycRJsKL7JKa96p
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||||||
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||||||
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KzJ7oRBkNRDQBeWLXk1x2AJJA4DJEoLqKCIdCZAsNlHQC5AML5EXGFFFFyJ0Iieg
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||||||
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v6t+JAnELoSnuUM1uRcFOVNkQL06lrvBZorOP8lBLwW+jjo3K1PyoOqGTlnmWHQ/
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6mIiymoiQxYiIx2nxQkium0wemEAswLnanCxXPXn1iiixnbROJIxjheZuOyhFmaj
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||||||
|
Kv3mGRtjivH89j/w/9/nW3TiQXQarjGTbiIXC2+aBuRqhulTHrjJIXLqYXerc3Yp
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||||||
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||||||
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||||||
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|
||||||
|
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|
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||||||
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|
||||||
|
|
||||||
|
kBEAA===
|
||||||
|
```
|
||||||
|
%%
|
||||||
Binary file not shown.
|
After Width: | Height: | Size: 21 KiB |
|
|
@ -0,0 +1,926 @@
|
||||||
|
---
|
||||||
|
|
||||||
|
excalidraw-plugin: parsed
|
||||||
|
tags: [excalidraw]
|
||||||
|
|
||||||
|
---
|
||||||
|
==⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠==
|
||||||
|
|
||||||
|
|
||||||
|
# Text Elements
|
||||||
|
%%
|
||||||
|
# Drawing
|
||||||
|
```compressed-json
|
||||||
|
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
|
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||||||
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||||||
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||||||
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||||||
|
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||||||
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||||||
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|
||||||
|
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||||||
|
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||||||
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||||||
|
|
||||||
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|
|
||||||
|
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||||||
|
|
||||||
|
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||||||
|
|
||||||
|
wwJjkIYKkKUM4Pf1OA4LUieFxFUK0IHgXwEPkMQy/k0PTyAKgIIMMIYNckALEPkL
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||||||
|
|
||||||
|
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||||||
|
|
||||||
|
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||||||
|
|
||||||
|
IzKhiMCkuEaLKPiJv3aJPwaIPjaP+B6I6L6MiJ3z+FaIGNGK6MGKaMyXGK/xGP6I
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||||||
|
|
||||||
|
mJmOqJaPmK/w+wtmEJnw71pEIH0HzHPAQAAAUe9mA+9lxVwEAnxQgoBWR9B9A1BF
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||||||
|
|
||||||
|
wDjgJVQ0AxJK9LkV1i0kQc8UcG8C9m8a0S8O8igABfFYEoMoCoCQZQWIeMToAAVQ
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||||||
|
|
||||||
|
2FmBcGcA4DhPwE6EOAADUABHAAMQACsAAVXAU0aYcQdAeYBARYcgZYaMNYNAOyJu
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||||||
|
|
||||||
|
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||||||
|
|
||||||
|
BlZhALRmMFQECkN0GCAQR0JkaUDkbkfkPkJAdcEUWoWMCUKUNkA0uUcgDgRUZUbI
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||||||
|
|
||||||
|
KAU0DULUF0N0KQI0U42kXUhAK0YgUEAMH0i0Z0PUcks0NkCoQsYQH0P0W0QEYMEU
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||||||
|
|
||||||
|
MMGsbUmMcUBMJMAoTvTMbMJcN8Q8aMy0ksA8Cse0KsPM1AfWffFqVMnsNsLELsBg
|
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|
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||||||
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||||||
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||||||
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|
||||||
|
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|
||||||
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||||||
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||||||
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||||||
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||||||
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|
||||||
|
AAwJCADAkQAA
|
||||||
|
```
|
||||||
|
%%
|
||||||
Binary file not shown.
|
After Width: | Height: | Size: 28 KiB |
|
|
@ -0,0 +1,554 @@
|
||||||
|
---
|
||||||
|
|
||||||
|
excalidraw-plugin: parsed
|
||||||
|
tags: [excalidraw]
|
||||||
|
|
||||||
|
---
|
||||||
|
==⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠==
|
||||||
|
|
||||||
|
|
||||||
|
# Text Elements
|
||||||
|
%%
|
||||||
|
# Drawing
|
||||||
|
```compressed-json
|
||||||
|
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||||||
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||||||
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||||||
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||||||
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||||||
|
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||||||
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||||||
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||||||
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aYA2qE4JXFzMtS6fZBEVZcDdWH+a+f+NhPDF1NRO8SZKac+KXMACjbDQ4lLWyO1H
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||||||
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|
||||||
|
BQLE7Z5GAv2CaN9fPWyNAmKfYW8Q4OYu46CVElleAzElKbEjAvE8nDNQvICKvSrX
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||||||
|
|
||||||
|
QhbNGF4mrIbLrAXHPWNJeb/dCY6fEuHCbARKXEbNk3bB5Dk73UqCU07KUlk5HbKO
|
||||||
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||||||
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||||||
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||||||
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|
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||||||
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|
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|
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||||||
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||||||
|
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||||||
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||||||
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||||||
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|
||||||
|
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||||||
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||||||
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||||||
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||||||
|
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||||||
|
|
||||||
|
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||||||
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|
||||||
|
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|
||||||
|
|
||||||
|
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|
||||||
|
|
||||||
|
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||||||
|
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||||||
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||||||
|
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||||||
|
|
||||||
|
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|
||||||
|
|
||||||
|
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||||||
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||||||
|
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||||||
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||||||
|
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||||||
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||||||
|
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||||||
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||||||
|
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||||||
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||||||
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||||||
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|
||||||
|
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||||||
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|
||||||
|
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||||||
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|
||||||
|
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||||||
|
|
||||||
|
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||||||
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|
||||||
|
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|
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|
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||||||
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||||||
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||||||
|
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|
||||||
|
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||||||
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||||||
|
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|
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||||||
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||||||
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||||||
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||||||
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|
||||||
|
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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|
||||||
|
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||||||
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||||||
|
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||||||
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||||||
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||||||
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||||||
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||||||
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|
||||||
|
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||||||
|
|
||||||
|
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||||||
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|
||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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|
||||||
|
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||||||
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|
||||||
|
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||||||
|
|
||||||
|
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||||||
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|
||||||
|
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||||||
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||||||
|
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||||||
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|
||||||
|
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||||||
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||||||
|
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||||||
|
|
||||||
|
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||||||
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|
||||||
|
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||||||
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|
||||||
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||||||
|
|
||||||
|
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||||||
|
|
||||||
|
CgQX3GgAHc2QLgCaAwQLYQ8+2OtgBEAcANwDKhQIBwBrMGoaQC8+95EIBQAr4Ffj
|
||||||
|
|
||||||
|
KhM+OAB7cI2jqBaSqAMADT4IANPhAAA=
|
||||||
|
```
|
||||||
|
%%
|
||||||
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|
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|
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Loading…
Reference in New Issue