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I ran into this problem when trying to do a project on one-electron reduced matrices of fermions. The math can be formulated as following:

Let $\{a_i\}_{i=1...m}$ be a set of variables with additional constraints

  1. $\sum_{i=1}^ma_i = N$,
  2. $a_i \in [0, 2]$.

Obviously, we also have $\text{min}(m) > N/2$.

So my question is that is there any way we could construct another set of variables, i.e. $\{b_i\}$, such that they could satisfy these constraints automatically?

For examples for constraint 2, we could let $b_i = 2 (1- \text{cos}(\theta_i))$, so that $\{b_i\}$ satisfy constraint 2 by construction.

Is there any way to do that for constraint 1 too?

L. Yu
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  • You could do something like $b_1=N-\sum_{i=\mathbf{2}}^m b_i$ and then choose the other $b_i$ in any preferred way. One should be cautious that in this way one might not obtain $b_1\in[0,2]$ though.

    You could also just take $b_i=N/m$ for all $i$

     – student91 Sep 30 '22 at 09:14

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