A How to check if a function doesn't depend on a variable?

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To determine if the function does not depend on a variable, it is essential to assess the independence of the variables involved, specifically ##x_i## and ##y_i##. If they are independent, one can evaluate whether the coefficient ##b## is approximately zero without considering ##a##. However, if there is a correlation between the variables, the value of ##a## must be taken into account to assess the significance of a non-zero ##b##. Given the inability to measure ##y_i##, regression analysis can be employed to check if the model ##z_i = a x_i## yields a statistically significant non-zero ##a##. Ultimately, the approach to testing the dependency hinges on the relationship between the variables and the available data.
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Hello! I have some experimental data points ##(z_i,dz_i)## and I know that in the most general case this variable can be written in terms of 2 other variables as ##z_i = ay_i+bx_i##. Beside ##z_i## I can also measure, for each point, ##x_i## (we can assume that the uncertainty in ##x_i## is negligible), but not ##y_i##. I suspect, based on some calculations, that (at least at the level of the experimental uncertainties, ##dz_i##) the ##bx_i## term will be negligible i.e. ##b\sim 0## given my uncertainties. Is there a way to test this experimentally, given my current data and the expected functional form? Thank you!
 
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An important question is whether the ##x_i##s and ##y_i##s are independent or are correlated. If they are independent, then you can consider whether ##b \approx 0## without regard to the value of ##a##. But if they are related, you must consider the value of ##a## to understand whether the addition of a nonzero ##b## is beneficial.
Since you can not measure the ##y_i##s, I am afraid that the best you can do is to use regression to determine if the model ##z_i = a x_i## has a statistically significant non-zero ##a##.
 
In Calculus, the Inverse/Implicit function theorems are usually used with this purpose, when given an expression in terms of ##x,y ##.
 
I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...

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