viscosity of the material that flows into the area vacated under the
rebounding continental plates.
- Using this technique, values around $\eta=10^{21} \text{Pa \thinmuskip s}
+ Using this technique, values around $\eta=10^{21} \text{Pa \; s}
= 10^{21} \frac{\text{N\; s}}{\text{m}^2}
= 10^{21} \frac{\text{kg}}{\text{m\; s}}$ have been found as the most
likely, though the error bar on this is at least one order of magnitude.
routines FEValues::shape_grad and FEValues::JxW
return
@f{align*}
-\text{FEValues::shape_grad}(i,l)&=D \mathbf x_K(p_l) G^{-1}(p_l)D(\varphi_i \circ \mathbf x_K)
+\text{FEValues::shape\_grad}(i,l)&=D \mathbf x_K(p_l) G^{-1}(p_l)D(\varphi_i \circ \mathbf x_K)
(p_l)
\\
\text{FEValues::JxW}(l) &= \sqrt{\det (G(p_l))} \ w_l.
*
* @f[
* r = r_{\text{inner}} + (r_\text{outer} - r_\text{inner})
- * \frac{1 - \tanh(\text{skewness}(1 - k/\text{n_shells}))}
+ * \frac{1 - \tanh(\text{skewness}(1 - k/\text{n\_shells}))}
* {\tanh(\text{skewness})}
* @f]
*
*
* This is based on the definition of elliptic coordinates $(u,v)$
* @f[
- * \left\lbrace\begin{align*}
+ * \left\lbrace\begin{aligned}
* x &= x_0 + c \cosh(u) \cos(v) \\
* y &= y_0 + c \sinh(u) \sin(v)
- * \end{align*}\right.
+ * \end{aligned}\right.
* @f]
* in which $(x_0,y_0)$ are coordinates of the center of the cartesian system.
*
*
* This class and classes derived from it are meant to build $Q$ and $R$
* matrices one row/column at a time, i.e., by growing $R$ matrix from an empty
- * $0\times0$ matrix to $N\timesN$, where $N$ is the number of added column
+ * $0\times 0$ matrix to $N\times N$, where $N$ is the number of added column
* vectors.
*
* As a consequence, matrices which have the same number of rows as each vector
* method for function minimization in which the Hessian matrix is only
* approximated. In particular, the Hessian is updated using the formula of
* Broyden, Fletcher, Goldfarb, and Shanno (BFGS):
- * \f[
+ * @f{align*}{
* H^{(k+1)} &= \left[
* I-\rho_{(k)} s^{(k)} \otimes y^{(k)}
* \right]
* y^{(k)} &\dealcoloneq g^{(k+1)} - g^{(k)} \\
* s^{(k)} &\dealcoloneq x^{(k+1)} - x^{(k)} \\
* \rho^{(k)} &\dealcoloneq \frac{1}{y^{(k)} \cdot s^{(k)}}
- * \f]
+ * @f}
* for a symmetric positive definite $H$. Limited memory variant is
* implemented via the two-loop recursion.
*