In social choice and voting theory, the profile of voters' rankings obviously determines all the head-to-head outcomes among n candidates. A theorem by McGarvey states that, given any head-to-head results you want, there is a profile of voters that will produce it. Very cool.

My question is... If you impose the condition that all voters unanimously prefer candidate A to candidate B, how does that limit the possible head-to-head results? Is anyone aware of any work done on this question?

I've come across an nxn matrix whose entry in row i and col j is min(i,j). For example, the 4x4 version would be
$\left( \begin{array}{cccc} 1 & 1 & 1 & 1\\ 1 & 2 & 2 & 2\\ 1 & 2 & 3 & 3\\ 1 & 2 & 3 & 4 \end{array} \right)$
Does anybody recognize this class of matrices? Does it have a name? I'd like to research what is known about it.

that there exist irrationals $$a$$ and $$b$$ such that $$a^{b}$$ is rational:
$$\sqrt{2}$$ is irrational. Let $$z=\sqrt{2}^{\sqrt{2}}$$. If $$z$$ is rational, the claim is proved. If not, then $$z^{\sqrt{2}}=2$$ proves the claim.

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