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After discussing the MYHockey algorithm with my uncle, I learned that one of the assumptions in my previous analysis is wrong. In particular, the rating associated with the strength of a team’s opponents is not computed from static values from a previous time, but they are updated with every game result.
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I recently move to Massachusetts where the youth sport of choice is ice hockey. I was watching my cousin play a game and, between the crashes of body checks into the boards that made me fear for my cousin’s life, my uncle told me about a website where he gets his intel on the other teams - myhockeyrankings.com.
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What should your thermostat do when you are away from home? Say you keep the temperature of your home, $x(t)$, at a fixed temperature $H_i$ when you are around. Assume that you are going to be leaving for a winter vacation at time $t=0$, and you want the home to be the same temperature when you return at time $t=T$, but you don’t care about the home’s temperature when you are gone. We will use Pontryagin’s maximum principle to achieve your desired final temperature, while minimizing the amount of energy used while you are gone.