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A Course on Cooperative Game Theory by Chakravarty et al. gives the simplest textual definitions I've seen on this. See page 164-165. Core: An assignment of houses to owners is in the core of the house exchange market if there does not exist a coalition $S$ of house owners that can redistribute the houses they own such that they all prefer the houses ...


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You didn't say your geographical area. In the United States, departments are typically not only willing to consider applicants without an MS--that most admitted PhD candidate have not had master's training is sort of the norm. As a consequence, the US degree process usually involve some number of years of course work (1 or 2 typically) with a ...


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If you're assuming a completely fair currency market, then it makes perfect sense: essentially what your equation is telling you is the rate at which the Xoxo is trading against itself. That is, if you take $x$ Xoxo's, you can trade it in for $z$ Zongs which you can then turn in for $y$ Yabbas which you can then turn in for $x$ Xoxo's. So your overall rate ...


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In order to compute the monthly interest, you know that in a year (which has 12 months) you get $6\%$ interest. Call $a$ the (decimal) monthly interest. Thus, $$ (1+a)^{12}=1.06. $$ Then you just solve the equation for $a$.


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If I have understood the construction of ${}^\ast\Bbb R$ properly, then the function defined as follows should work. $$u(f)_n=\sum_{k=0}^nf(k)$$ Here $u(f)_n$ denotes the $n$-th member of the hyperreal $u(f)$. If $f\prec g$, then $u(f)_n<u(g)_n$ for all $n\ge m$. There are cofinitely many $n$ greater than $m$, so regardless of the choice of ultrafilter, ...



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