How to Extract Parameters from Non-Invertible Functions Using Experimental Data?

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SUMMARY

The discussion focuses on extracting parameters \(\omega\), \(T\), and \(x\) from non-invertible functions related to energy \(E\) and angular momentum \(J\) of mesons, specifically using experimental data. The equations provided are \(E = \frac{2T}{\omega}\left(\arcsin x + \frac{1}{x}\sqrt{1-x^2}\right)\) and \(J = \frac{T}{\omega^2}\left(\arcsin x + x \sqrt{1-x^2}\right)\). The challenge arises from the non-invertibility of these functions, complicating the extraction process. Participants are seeking methods to effectively utilize the given data to derive the parameters for each meson.

PREREQUISITES
  • Understanding of String Theory concepts
  • Familiarity with curve fitting techniques
  • Knowledge of inverse functions and their properties
  • Proficiency in mathematical modeling and data analysis
NEXT STEPS
  • Research numerical methods for parameter extraction from non-invertible functions
  • Explore optimization techniques for fitting experimental data to theoretical models
  • Learn about the use of software tools like MATLAB or Python for curve fitting
  • Investigate the application of Monte Carlo methods in parameter estimation
USEFUL FOR

Researchers in theoretical physics, particularly those working on String Theory, data analysts involved in experimental physics, and anyone engaged in mathematical modeling of physical phenomena.

Sebastian
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Homework Statement



This is for a project in String Theory, but it's actually a curve fitting question. I've derived the two equations given below, for the energy E and angular momentum J of a meson as a function of some quantities \omega, T, x. Now, I have experimental data for E as a function of J for each meson (for example, for the rho meson E(1) = 775.5 MeV, E(2) = 1318.3 MeV, etc.). I need to use the data in order to extract \omega, T, x for each meson. Since the functions I found are not invertible, I don't know how to continue from here. Any help would be appreciated :)

Homework Equations



E = \frac{2T}{\omega}\left(\arcsin x + \frac{1}{x}\sqrt{1-x^2}\right)
J = \frac{T}{\omega^2}\left(\arcsin x + x \sqrt{1-x^2}\right)

The Attempt at a Solution



See above.
 
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