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calculations. This commit changes results (but not visibly)
for the l_avg_Lscale code, but since this code is not the default
in CLUBB, the code remains bit-for-bit since the last commit.
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Notebooks for proving formulas (#1151)

* Implement equations 1 through 5.

* Update TrivariateGaussian.ipynb

Adds parentheses to delta equation

* Fixed sigma equation.

* Type in higher order equation.

* Implement rhs of the equation (48)

* Implement ith Gaussian

* Somehow add integration with sympy

* Somehow add integration with sympy

* Added integration with sympy and adjusted variables.

* Integration now works.

* Add 4th and second order moments.

* check equation 56 with 20, 11 and 4

* check equation 56 with 20, 11 and 4

* remove unnecessary symbols

* Move documents and rename

* Rename document

* Add files and start to implement the analytic integral

* Add functions

* Add symbols and functions.
Tried analytic integration with simpler functions but the triple integral still does not work. For integrating numerically I like to use lambdify and then numpy but therefore the symbols have to be renamed because lambdify does not like backslashes.

* Numerical evaluation of the first, second, third, and fourth order moments of the triple Gaussian with limits -10 to 10.

* Added mixture fractions to the Gaussian.

* Convert results to dataframe for better readability.

* Check results for third Gaussian, also adjusted the third Gaussian.

* Tidy up code.

* Changed name of file.
Added function for evaluating the fraction  w_prime_4_bar divided by w_prime_2_bar_squared which is parameterized.

* Added columns to the dataframe to check if the numerical evaluation of the 4th order moment is equal to the parameterized evaluation of the 4th order moment.

* Adjusted symbols and added the constant for lambda_w.

* Added documentation.

* Added check for w_prime_4_bar calculated just with the parameters.

* Improved functions and improved calculating the moments from a dataframe.

* Tidied up code

* Tidied up code

* Reordered document

* Added some files and restructured the document.

* Added a file to compute another function

* Added a file for computing the 4th order moment based on the parameters and checking this with the numeric integrals.

* fixed notebook

* Checked some equations analytically.

* Added document for checking eq_26

* Added document for checking eq_24

* Added document for checking eq_33

* Fixed some parameters

* finished document for computing the moments numerically.

* Verified(?) eq 75

* Changed values for ?verifying? eq 75

* Adjusted alphas and deltas but did not run it to the end yet.

* Ran the code with the adjusted alphas and deltas.

* Calculated the integral based on the marginal of G_3

* Ran the notebook again.

* Added Checking for eq 80. Not done yet.

* Added new document.

* Calculated w_bar

* Finished 1 dim integral.

* Added calculation for sigma_lambda_w and lambda_w.

* Added calculation for sigma_lambda_w and lambda_w.

* Added third Gaussian.

* Calculated 4th order moment.

* Added files and split up evaluation of the moments for w

* Verified all moments for w.

* Fixed a typo.

* Set up document for another integral.

* Working on the integral for w_prime_theta_l_prime_bar.

* Working on the integral for w_prime_theta_l_prime_bar.

* Did the check for w_prime_theta_l_prime_bar.

* Did the check for w_prime_theta_l_prime_bar.

* Did the check for w_prime_theta_l_prime_bar with rational numbers.

* Add numerical difference.

* Add numerical difference.

* Finished the check for w_prime_theta_l_prime

* Finished the check for w_prime_squared_theta_l_prime

* Finished the check for theta_l_prime_3_bar

* Finished the check for w_prime_theta_l_prime_squared_bar

* Did the checks again with arbitrary distinct values.

* Did the checks again with arbitrary distinct values.

* Add check for theta_l_prime_2_bar.

* Calculated $\beta$.

* Corrected a minor mistake.

* Backed out beta from the theta_prime_l_bar equation.

* Checked the beta equation numerically.

* Printed all NaN values.

* Reformat Check.

* Reformat Check.

* Reformat Check.

* Reformat Check.

* Reformat Check.

* Reformat Check.

* Reformat Check.

* Reformat Check.

* Reformat Check.

* Changed Heading.

* Changed the limits of integration.

* Delete doc/.ipynb_checkpoints directory

* Rearrange symbols

* Added beta to the dataframe.

* Added diagnostic values to the dataframe.

* Fixed typo.

* rearranged dataframe

* rearranged dataframe

* rearranged dataframe

* rearranged dataframe

* rearranged dataframe

* Computed the triple integral again.

* Rearranged symbols.

* Working on the trivariate Normal.

* Fixed numerical integration for triple gaussian

* plot relationship between delta and lambda_w

* plotted proposed relationship

* Rearranged files and computed everything again.

* Added README.md

* Added file to README.md

* Add function for plotting

* Finished numerical check for w_prime_q_t_prime_theta_l_prime_bar_beta.ipynb

* Finished numerical check for w_prime_q_t_prime_theta_l_prime_bar_beta.ipynb

* Finished numerical check for theta_l_prime_q_t_prime_bar.ipynb

* Worked on plotting

* Worked on README.md

* replace q_t with r_t

* replace q_t with r_t

* replace q_t with r_t

* Add explanation

* Reformat code

* Corrected a file

* Corrected a file

* Worked on checking w_prime_r_t_prime_theta_l_prime_bar_beta.ipynb

* Simplified creating of dataframe.

* - Added calculation for w_prime_r_t_prime_theta_l_prime_bar_E.ipynb and improved diagnostics output.

* Changed all sigma_lamda to sigma_3 and plotted \lambda vs \delta

* Plotted \lambda vs \delta with the "initial condition"

* Plotted \lambda vs \delta with the "initial condition"

* Update README.md

* Reran jupyter notebooks

* Refactored code

* Reran notebook

* Rearranged README.md

* Rearranged README.md

* Remove files

---------

Co-authored-by: Vincent Larson &lt;vlarson@users.noreply.github.com&gt;
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