Commit fe183f70 authored by Charles G Waldman's avatar Charles G Waldman
Browse files

align -> aligned (for PDF output)

parent 9bb8da6b

docs/source_docs/references/size_distributions.rst

+10 −10
Original line number Diff line number Diff line
@@ -233,20 +233,20 @@ The mean of this distribution is computed as 
.. math::

   :label: eq_normal_dist_FN_to_FV_mean



   \begin{align}

   \begin{aligned}

       \mu_V =& \frac{ \int_{-\infty}^{\infty} x \tilde{f}_X^V(x) dx }{\int_{-\infty}^{\infty} \tilde{f}_X^V(x) dx } \\[10pt]

             =& \frac{3\sigma_N^4 + 6\sigma_N^2\mu_N^2 + \mu_N^4}{3\sigma_N^2 \mu_N + \mu_N^3}

   \end{align}

   \end{aligned}



and the variance is given by



.. math::

   :label: eq_normal_dist_FN_to_FV_variance



   \begin{align}

   \begin{aligned}

       \sigma^2_V =& \frac{ \int_{-\infty}^{\infty} (x - \mu_V)^2 \tilde{f}_X^V(x) dx }{\int_{-\infty}^{\infty} \tilde{f}_X^V(x) dx } \\[10pt]

                  =& \frac{3\sigma_N^4 (3\mu_N + 2b) + \sigma^2(\mu_N^3 + 6\mu_N^2 b + 3\mu_Nb^2)+\mu_N^3 b^2}{3\sigma_N^2 \mu_N + \mu_N^3}

   \end{align}

   \end{aligned}



where :math:`b = (\mu_N - \mu_V)`. Because the volume-weighted distribution is normally distributed, the computed

mean and variance can be substituted directly into :eq:`eq_normal_dist`.
@@ -264,10 +264,10 @@ mean and standard deviation, :math:`\mu_V` and :math:`\sigma_V`, by solving the 


.. math::



   \begin{align}

   \begin{aligned}

     f_1(\mu_N,\sigma_N) &= \left( 3\sigma_N^4 + 6\sigma_N^2\mu_N^2 + \mu_N^4 \right) - \mu_V \left( 3\sigma_N^2 \mu_N + \mu_N^3 \right) = 0 \\

     f_2(\mu_N,\sigma_N) &= \left(3\sigma_N^4 (3\mu_N + 2b) + \sigma^2(\mu_N^3 + 6\mu_N^2 b + 3\mu_Nb^2)+\mu_N^3 b^2\right) - \sigma^2_V \left( 3\sigma_N^2 \mu_N + \mu_N^3 \right) = 0

   \end{align}

   \end{aligned}



This system is solved using the homotopy method outlined in :cite:t:`Burden10`. To ensure rapid convergence, initial guesses for

:math:`\mu_N` and :math:`\sigma_N` are calculated from the density function created by dividing :eq:`eq_normal_dist` by :math:`x^3`
@@ -288,11 +288,11 @@ are found by setting the derivative of :eq:`eq_normal_dist_FV_to_FN_initial_expr 
.. math::

   :label: eq_normal_dist_FV_to_FN_discrete_values



   \begin{align}

   \begin{aligned}

     x_{\mathrm{min}} \; =& \; \left( \mu_V - \sqrt{\mu_V^2 -12\sigma_V^2}\right)/2 \\

     x_{\mathrm{mid}} \; =& \; \left(\mu_V + \sqrt{\mu_V^2 -12\sigma_V^2}\right)/2 \\

     x_{\mathrm{max}} \; =& \; 2x_{\mathrm{mid}} - x_{\mathrm{min}}

   \end{align}

   \end{aligned}



.. _fig_normal-dist-pdf-approx_FN:
@@ -464,10 +464,10 @@ while the parameter :math:`\mu` is computed directly from the other distribution 


.. math::



   \begin{align}

   \begin{aligned}

       \mu_V =& \mu_N + 3.0\sigma^2 \\[10pt]

       \mu_N =& \mu_V - 3.0\sigma^2

   \end{align}

   \end{aligned}



.. rubric:: Sampling a log-normal distribution