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Eigenranking
My team matrix: [
[0,3,3,2,2],
[4,0,1,3,1],
[1,4,0,4,2],
[2,1,3,0,2],
[3,1,4,2,0]
}
Team 3: rating = 0.4848807961654844 Team 5: rating = 0.47317429197368976 Team 1: rating = 0.4632070845269747 Team 2: rating = 0.4163867448282543 Team 4: rating = 0.39122624975897735
Team 3 had the highest rating at 0.48488.
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Modeling a steady state heat distribution in 2D
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Modeling 2D Heat Flow
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Random heat evolution problem
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Steady state heat flow with source
My heat flow problem where I found the steady state temperature distribution: $$u_t=3u_{xx}+1$$
My boundary conditions were: $$u(0,t)=-3$$ and $$u(1,t)=4$$
Since $u_t=0$ when the bar is in steady state I used this to solve for $u_{xx}$, which I got was $u_{xx}=\frac{-1}{3}$. Then integrating twice: $$u(x)=\frac{-1}{6}x^2+Cx+D$$
I found these constants by using my boundary conditions: $C=\frac{43}{6}$ and $D=-3$
Finally I got: $$u(x)=\frac{-1}{6}x^2+\frac{43}{6}x-3$$