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A random vibration problem
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Steady state heat flow with source
Heat flow with a constant internal heat source is governed by
$$u_t = 2u_{xx} + 3, u(0,t)=-4, u(1,t)=3.$$
Find the steady state temperature distribution.
Steady state occurs when $u_t=0$. Solving the above equation for $u_{xx}$ given $u_t=0$, yields $u_{xx}=-3/2$. Integrating this twice gives $u(x)=\frac{-3}{4}x^2+cx+d.$ Using the boundary condition to solve for c and d gives you the steady state equation of $$u(x)=\frac{-3}{4}x^2+\frac{31}{4}x-4$$
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Random heat evolution problem
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Eigenranking
My matrix was:
matrix = [
[0,2,1,2,4],
[4,0,4,4,2],
[4,2,0,1,2],
[3,2,4,0,4],
[4,1,4,4,0]
]
The ranking for my league is:
Team 2: rating = 0.5165517842331686 Team 4: rating = 0.48670829470439997 Team 5: rating = 0.47767125643216296 Team 1: rating = 0.3784150823149017 Team 3: rating = 0.3534423380504801
Team 2 is the best.
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Modeling 2D Heat Flow
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Modeling a steady state heat distribution in 2D