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The issue that I am dealing with now ends up with the solution of a second order equation. The solutions are the Z positions of a point in 3D. So, basically I have two points with the Z positions of $Z_i (i=1,2)$.

To choose between these two points I need to define the equation $|Z_i-C|$ where $C$ is a constant. The appropriate $Z_i$ is the one that minimizes this equation ($|Z_i-C|$).

Now I am wondering if there is anyway to mathematically represent this procedure in an equation-form. By equation-form I mean something like $Z= min ( ... |Z_i-C| ... $.

Thank you

NESHOM
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  • I'm very sorry but I can't understand what you intend. Maybe you can add some more context. While you're at it you may want to use MathJax to typeset mathematical formulae correctly. – AlexR Jan 19 '15 at 18:18
  • Thanks for your reply. I just modified my question and tried to make it clear. – NESHOM Jan 19 '15 at 18:26

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What you are looking for is called the $\mathrm{arg\;min}$ operator: $$Z = \mathop{\rm arg\;min}_{Z\in\{Z_1,Z_2\}} |Z-C|$$

AlexR
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  • perfect, thank you and sorry for this foolish question. – NESHOM Jan 19 '15 at 18:30
  • @NESHOM There's nothing foolish about asking for a notation :) This one is most common in optimization theory btw. – AlexR Jan 19 '15 at 18:31
  • Is it also correct of I write something like this for example? $Z = \mathop{\rm arg;min}{i\in{1,...,10}} |{ }^{C_j}\textrm{Z}_i-C|$ – NESHOM Jan 19 '15 at 18:35
  • @NESHOM The $\mathrm{arg;min}$ in this case would give you the index $i$. To fix this write $$i_0 := \mathop{\rm arg;min}{i\in{1,\ldots,10}} |^{C_j}Z_i - C|\Z := Z{i_0}$$(or index the whole construct like $Z_{\ldots}$, but that's a mess^^) – AlexR Jan 19 '15 at 18:36
  • So, if instead of two Z ($Z_1,Z_2$) there was 10 of them, how should we shorten it and not include all of the Z? – NESHOM Jan 19 '15 at 18:38
  • Is there any modified version of what you suggested originally for more than two Z? – NESHOM Jan 19 '15 at 18:39
  • @NESHOM You can use set-builder notation there:$$Z = \mathop{\rm arg;min}_{{Z_i : i=1..10}} \ldots$$ – AlexR Jan 19 '15 at 18:39
  • Great, thank you very much :) – NESHOM Jan 19 '15 at 18:40