https://r2.erweima.ai/mathbot/image/17247446037475s4zl24o.jpg

To solve the minimization problem described in the image, you need to follow these steps:

1. **Objective Function**: The minimization problem involves minimizing the sum of squared differences between the detected signal \( S_i \) and the theoretical signal \( S_i(T, \epsilon, \lambda) \) across multiple wavelengths \( \lambda_i \). Mathematically:

   \[
   \min_{T, \epsilon} \sum_{i=1}^{N} \left[ \frac{S_i - S_i(T, \epsilon, \lambda_i)}{\sigma_i} \right]^2
   \]

   The parameters \( T \) (temperature) and \( \epsilon(\lambda, T) \) (emissivity model) are to be determined.

2. **Emissivity Model**: The emissivity \( \epsilon(\lambda) \) is modeled using a second-order polynomial function for metals:

   \[
   \epsilon(\lambda) = a_0 + a_1 \lambda + a_2 \lambda^2 + \dots
   \]

   In the case of liquid pure Ti metal, the emissivity is approximated as a linearly decreasing function of wavelength:

   \[
   \epsilon(\lambda_i) = B - A \frac{\lambda_i - \lambda_1}{\lambda_8 - \lambda_1}
   \]

   Here, \( A \) and \( B \) are constants to be determined.

3. **Minimization Strategy**: The goal is to find the values of \( T \), \( A \), and \( B \) that minimize the objective function. This can be achieved using numerical optimization techniques such as:

   - **Gradient Descent**: Use iterative methods to adjust the parameters \( T \), \( A \), and \( B \) to reduce the sum of squared differences.
   - **Levenberg-Marquardt Algorithm**: A common choice for non-linear least squares problems, especially when dealing with models like these.

4. **Parameter Fitting**: You apply the chosen optimization technique to fit the parameters \( T \), \( A \), and \( B \) such that the residuals (the differences between the detected and theoretical signals) are minimized. The least-squares fitting algorithm would adjust the parameters iteratively until convergence is reached.

5. **Implementation**: This can be implemented in software tools like Python (using libraries such as SciPy's `curve_fit` or `minimize`), MATLAB, or other numerical analysis environments.

Would you like more details on implementing this algorithm in Python, or do you have specific questions?

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### Related Questions
1. How does the Levenberg-Marquardt algorithm work for non-linear least squares?
2. What are the typical initial guesses for \( T \), \( A \), and \( B \) in such minimization problems?
3. How can numerical stability issues be addressed when fitting these types of models?
4. What is the role of the noise term \( \sigma_i \) in the optimization process?
5. How can we extend the model for more complex emissivity functions beyond linear approximations?

### Tip
Always start with good initial guesses for the parameters when performing non-linear optimization to avoid local minima.

Explore how to solve a minimization problem using numerical optimization techniques. Learn about the objective function, parameter fitting, and the Levenberg-Marquardt algorithm for non-linear least squares. Ideal for researchers and practitioners in computational sciences.

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