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Not an essay or book, but there is a rather informative and clear video address by Timothy Gowers (a Fields Medalist) in 2000 on the "Importance of Mathematics" for an audience containing non-mathematicians in Paris. He specifically addresses the issue of the (enormous) benefits that mathematics yields on a (very small) investment, its intrinsic cultural ...


2

Probably because: The utility of the lowest type is being normalized to zero $u(t,\underline v)=0$ ? $G(\overline v |t)=1 \,\forall t$ ? $G(\underline v |t)=0\, \forall t$ ? But it is hard to say because the question did not define what $G$ is (I am assuming is a CDF with support on $[\underline v, \overline v]$).


1

You can use the following function for calculating floor: $$f(x)=x-\frac{1}{2}-\frac{\arcsin(\sin(\pi(x-\frac{1}{2})))}{\pi}$$ Then, for your purpose, you can simply use: $$\frac{f(xN)}{N-1}$$ If it helps, then you can also use the following function for calculating ceiling: $$f(x)=x+\frac{1}{2}+\frac{\arctan(\tan(\pi(-x-\frac{1}{2})))}{\pi}$$ If ...



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