attempting align -> aligned fix

docs/guide/chunkgraphbvps.rst

@@ -69,12 +69,12 @@ the following PDE:
 
 .. math::
 
-   \begin{align*}
+   \begin{aligned}
    (\Delta + k^2) u^{\textrm{scat}} &= 0 & \textrm{ in } \mathbb{R}^2 \setminus \Omega \; , \\
    \frac{\partial u^{\textrm{scat}}}{\partial n} &= -\frac{\partial u^{\textrm{inc}}}{\partial n} & \textrm{ on } \Gamma \; , \\
    \sqrt{|x|} \left( \frac{x}{|x|} \cdot \nabla u^{\textrm{scat}} - ik u^{\textrm{scat}} \right )
    &\to 0 & \textrm{ as } |x|\to \infty \; ,
-   \end{align*}
+   \end{aligned}
 
 The Green function for the Helmholtz equation is
 
@@ -87,10 +87,10 @@ single and double layer potential operators
 
 .. math::
 
-   \begin{align*}
+   \begin{aligned}
    [S_{k}\sigma](x) &:= \int_\Gamma G_k(x,y) \sigma(y) ds(y)  \\
    [D_{k}\sigma](x) &:= \int_\Gamma n(y)\cdot \nabla_y G_k(x,y) \sigma(y) ds(y) 
-   \end{align*}
+   \end{aligned}
 
 A robust choice for the layer potential representation for this problem is
 a *right preconditioned combined field* layer potential, which is a linear combination
@@ -112,10 +112,10 @@ results in the equation
 
 .. math::
 
-   \begin{align*}
+   \begin{aligned}
    -\frac{\partial u^{\textrm{inc}}(x_0)}{\partial n} &= \lim_{x\in \mathbb{R}^2\setminus \Omega, x\to x_0}  \beta[(S_{k} + i\alpha D_{k} S_{ik})\sigma](x) \\
    &= \sigma(x_0) + \beta[S_{k}'\sigma ](x_0) + i\beta \alpha [(D_{k}' - D_{ik}')S_{ik}\sigma](x_{0}) + i \beta \alpha [S_{ik}'S_{ik}'\sigma](x_0) \;,
-   \end{align*}
+   \end{aligned}
 
 where $S_{k}'$ and $D_{k}'$ represent the normal derivatives of 
 the single and double layer potentials respectively, and we have used 
@@ -186,11 +186,11 @@ for the potential $u$ on a domain $\Omega$ with boundary $\Gamma$,
 .. math::
 
 
-   \begin{align*}
+   \begin{aligned}
    \Delta u &= 0 & \textrm{ in } \Omega \; , \\
    u &= f_{D} & \textrm{ on } \Gamma_{D} \; , \\
    \frac{\partial u}{\partial n} &= f_{N} & \textrm{ on } \Gamma_{N} \; , 
-   \end{align*}
+   \end{aligned}
 
 where $\Gamma_{D}$ and $\Gamma_{N}$ are a partition of the boundary, 
 i.e. $\Gamma = \Gamma_{D} \cup \Gamma_{N}$, with 
@@ -223,20 +223,20 @@ where
 
 .. math::
 
-   \begin{align*}
+   \begin{aligned}
    [S_{\Gamma_{N}}\sigma_{N}](x) = \int_{\Gamma_{N}} G_0(x,y) \sigma_{N}(y) ds(y) \; ,\\
    [D_{\Gamma_{D}}\sigma_{D}](x) = \int_{\Gamma_{D}} n(y) \cdot \nabla_{y} G_0(x,y) \sigma_{D}(y) ds(y) \; .
-   \end{align*}
+   \end{aligned}
 
 On imposing the boundary conditions, we get the following system of integral equations
 for $\sigma_{D}$ and $\sigma_{N}$
 
 .. math::
 
-   \begin{align*}
+   \begin{aligned}
    \sigma_{D}(x) - 2[D_{\Gamma_{D}}\sigma_{D}](x) + 2[S_{\Gamma_{N}}\sigma_{N}](x) &= f_{D} \,, \,\, \textrm{ on } \Gamma_{D} \,, \\
    \sigma_{N}(x) - 2[D'_{\Gamma_{D}}\sigma_{D}](x) + 2[S'_{\Gamma_{N}}\sigma_{N}](x) &= f_{N} \,,\,\,  \textrm{ on } \Gamma_{N} \,, 
-   \end{align*}
+   \end{aligned}
 
 where $S'$ and $D'$ are restrictions of the normal derivatives of 
 $S$ and $D$ respectively. 

docs/guide/simplebvps.rst

@@ -26,10 +26,10 @@ value problem
 
 .. math::
 
-   \begin{align*}
+   \begin{aligned}
    \mathcal{L} u &= 0 & \textrm{ in } \Omega \; ,\\
    \mathcal{B} u &= f & \textrm{ on } \Gamma \; ,
-   \end{align*}
+   \end{aligned}
 
 where :math:`\mathcal{L}` is a linear, constant coefficient PDE
 operator and :math:`\mathcal{B}` is a linear trace operator that
@@ -87,10 +87,10 @@ Consider the Laplace Neumann boundary value problem:
 
 .. math::
 
-   \begin{align*}
+   \begin{aligned}
    \Delta u &= 0 & \textrm{ in } \Omega \; ,\\
    \frac{\partial u}{\partial n} &= f & \textrm{ on } \Gamma \; ,
-   \end{align*}
+   \end{aligned}
 
 where :math:`n` is the outward normal.
 This is a model for the equilibrium heat distribution in a
@@ -133,14 +133,14 @@ results in the equation
 
 .. math::
 
-   \begin{align*}
+   \begin{aligned}
    f(x_0) &= \lim_{x\in \Omega, x\to x_0} n(x_0)\cdot \nabla_x
    \int_\Gamma G_0(x,y) \sigma(y) \, ds(y)  \\
    &= \frac{1}{2} \sigma(x_0) + P.V. \int_\Gamma n(x_0) \cdot \nabla_x G_0(x_0,y)
    \sigma(y) \, ds(y) \\
    &=: \left [\left ( \frac{1}{2} \mathcal{I} + \mathcal{S}' \right ) \sigma
    \right ] (x_0) \; ,
-   \end{align*} 
+   \end{aligned} 
 
 where :math:`P.V.` indicates that the integral should be interpreted
 in the principal value sense, :math:`\mathcal{I}` is the identity operator, and
@@ -245,12 +245,12 @@ in the exterior of the object:
 
 .. math::
 
-   \begin{align*}
+   \begin{aligned}
    (\Delta + k^2) u^{\textrm{scat}} &= 0 & \textrm{ in } \mathbb{R}^2 \setminus \Omega \; , \\
    u^{\textrm{scat}} &= -u^{\textrm{inc}} & \textrm{ on } \Gamma \; , \\
    \sqrt{|x|} \left( \frac{x}{|x|} \cdot \nabla u^{\textrm{scat}} - ik u^{\textrm{scat}} \right )
    &\to 0 & \textrm{ as } |x|\to \infty \; ,
-   \end{align*}
+   \end{aligned}
 
 The Green function for the Helmholtz equation is
 
@@ -263,10 +263,10 @@ single and double layer potential operators
 
 .. math::
 
-   \begin{align*}
+   \begin{aligned}
    [S\sigma](x) &:= \int_\Gamma G_k(x,y) \sigma(y) ds(y)  \\
    [D\sigma](x) &:= \int_\Gamma n(y)\cdot \nabla_y G_k(x,y) \sigma(y) ds(y) 
-   \end{align*}
+   \end{aligned}
 
 A robust choice for the layer potential representation for this problem is
 a *combined field* layer potential, which is a linear combination
@@ -281,10 +281,10 @@ results in the equation
 
 .. math::
 
-   \begin{align*}
+   \begin{aligned}
    -u^{\textrm{inc}}(x_0) &= \lim_{x\in \mathbb{R}^2\setminus \Omega, x\to x_0} [(D-ik S)\sigma](x)  \\
    &= \frac{1}{2} \sigma(x_0) + [ (\mathcal{D} -ik \mathcal{S})\sigma ](x_0)
-   \end{align*}
+   \end{aligned}
 
 where we have used the exterior jump condition for the double layer potential.
 As above, the integrals in the operators restricted to the boundary must sometimes

docs/guide/transmission.rst

@@ -44,13 +44,13 @@ the following transmission boundary value problem
 
 .. math::
 
-   \begin{align*}
+   \begin{aligned}
    \Delta u_{j} + k_{j}^2 u_{j} &=0 & \textrm{ in } \Omega_{j} \; , j=1,2\ldots n_{r}\\
    u_{r_{\ell,+}} - u_{r_{\ell,-}} &= f_{\ell} & \textrm{ on } \Gamma_{\ell} \;, \ell=1,2,\ldots n_{e}\,, \\
    \beta_{r_{\ell,+}}\frac{\partial u_{r_{\ell,+}}}{\partial n} - \beta_{r_{\ell,-}}\frac{\partial u_{r_{\ell,-}}}{\partial n} &= g_{\ell} & \textrm{ on } \Gamma_{\ell}\;, \ell=1,2,\ldots n_{e}\,,\\
    \sqrt{|x|} \left( \frac{x}{|x|} \cdot \nabla u_{1} - ik_{1} u_{1} \right )
    &\to 0 & \textrm{ as } |x|\to \infty \;.
-   \end{align*}
+   \end{aligned}
 
 Here $\beta_{j} = 1$ for the transverse magnetic mode and 
 $\beta_{j} = 1/\varepsilon_{j}$ for the transverse electric mode. 
@@ -73,10 +73,10 @@ Recall that the single and and double layer potentials are given by
 
 .. math::
 
-   \begin{align*}
+   \begin{aligned}
    [S_{k}\sigma](x) &:= \int_\Gamma G_k(x,y) \sigma(y) ds(y)\;,  \\
    [D_{k}\sigma](x) &:= \int_\Gamma n(y)\cdot \nabla_y G_k(x,y) \sigma(y) ds(y) \;,
-   \end{align*}
+   \end{aligned}
 
 where $\Gamma = \cup_{\ell=1}^{n_{e}} \Gamma_{\ell}$.
 
@@ -85,16 +85,16 @@ and double layer potential operators as:
 
 .. math::
 
-   \begin{align*}
+   \begin{aligned}
    u_{j}(x) &= \frac{1}{\beta_{j}}[D_{k_{j}}\sigma](x) - [S_{k_{j}}\mu](x) & \textrm{ in } \Omega_{j}\;,
-   \end{align*}
+   \end{aligned}
 
 Then, imposing the boundary conditions on this representation results
 in the following equation for $\sigma, \mu$:
 
 .. math::
 
-   \begin{align*}
+   \begin{aligned}
    \begin{bmatrix}
    \sigma\\
    \mu
@@ -113,7 +113,7 @@ in the following equation for $\sigma, \mu$:
    \beta_{r_{\ell,+}} \beta_{r_{\ell,-}}f_{\ell}\\
    g_{\ell}
    \end{bmatrix} \;\; \textrm{ on } \Gamma_{\ell}\;, 
-   \end{align*}
+   \end{aligned}
 
 for $\ell=1,2,\ldots n_{e}$ where $\gamma_{\ell} = \frac{2}{\beta_{r_{\ell,+}} + \beta_{r_{\ell,-}}}$.