Latex fix

Author: WardBrian

src/reference-manual/transforms.qmd

@@ -477,12 +477,12 @@ $$
 For the transform, Stan uses the first part of an isometric log ratio
 transform; see [@egozcue+etal:2003] for the basic definitions and
 Chapter 3 of [@filzmoser+etal:2018] for the pivot coordinate version
-used here.  Stan uses the isometric log ratio transform because it 
+used here.  Stan uses the isometric log ratio transform because it
 results in equal variances of the the constrained sum to zero
 vector see, e.g.,[@seyboldt:2024]. Simpler alternatives, such as setting the
 final element to the negative sum of the first elements, do not result in
-equal variances. The $N - 1$ unconstrained parameters are independent, however, 
-the sum-to-zero constraint induces a negative correlation across the 
+equal variances. The $N - 1$ unconstrained parameters are independent, however,
+the sum-to-zero constraint induces a negative correlation across the
 constrained vector values.
 
 ### Zero sum transform {-}
@@ -520,7 +520,7 @@ It maps an unconstrained vector $y \in \mathbb{R}^N$ to a zero-sum vector $x \in
 $$
 \sum_{n=1}^{N + 1} x_n = 0.
 $$
-The values are defined inductively, starting with 
+The values are defined inductively, starting with
 $$
 x_1 = \sum_{n=1}^N \frac{y_n}{\sqrt{n \cdot (n + 1)}}
 $$
@@ -536,7 +536,7 @@ $$
 \sum_{n = 1}^{N + 1} x_n = 0
 $$
 by construction, because each of the terms added to $x_{n}$ is then
-subtracted from $x_{n + 1}$ the number of times it shows up in earlier terms. 
+subtracted from $x_{n + 1}$ the number of times it shows up in earlier terms.
 
 ### Absolute Jacobian determinant of the zero sum inverse transform {-}
 
@@ -757,14 +757,14 @@ $$
 ## Stochastic Matrix {#stochastic-matrix-transform.section}
 
 The `column_stochastic_matrix[N, M]` and `row_stochastic_matrix[N, M]` type in
-Stan represents an \(N \times M\) matrix where each column (row) is a unit simplex
-of dimension \(N\). In other words, each column (row) of the matrix is a vector
+Stan represents an $N \times M$ matrix where each column (row) is a unit simplex
+of dimension $N$. In other words, each column (row) of the matrix is a vector
 constrained to have non-negative entries that sum to one.
 
 ### Definition of a Stochastic Matrix {-}
 
-A column stochastic matrix \(X \in \mathbb{R}^{N \times M}\) is defined such
-that each column is a simplex. For column \(m\) (where \(1 \leq m \leq M\)):
+A column stochastic matrix $X \in \mathbb{R}^{N \times M}$ is defined such
+that each column is a simplex. For column $m$ (where $1 \leq m \leq M$):
 
 $$
 X_{n, m} \geq 0 \quad \text{for } 1 \leq n \leq N,
@@ -790,8 +790,8 @@ $$
 \sum_{m=1}^N X_{n, m} = 1.
 $$
 
-This definition ensures that each column (row) of the matrix \(X\) lies on the
-\(N-1\) dimensional unit simplex, similar to the `simplex[N]` type, but
+This definition ensures that each column (row) of the matrix $X$ lies on the
+$N-1$ dimensional unit simplex, similar to the `simplex[N]` type, but
 extended across multiple columns(rows).
 
 ### Inverse Transform for Stochastic Matrix {-}
@@ -801,8 +801,8 @@ as simplex, but applied to each column (row).
 
 ### Absolute Jacobian Determinant for the Inverse Transform {-}
 
-The Jacobian determinant of the inverse transform for each column \(m\) in
-the matrix is given by the product of the diagonal entries \(J_{n, m}\) of
+The Jacobian determinant of the inverse transform for each column $m$ in
+the matrix is given by the product of the diagonal entries $J_{n, m}$ of
 the lower-triangular Jacobian matrix. This determinant is calculated as:
 
 $$

src/reference-manual/types.qmd

@@ -678,7 +678,7 @@ priors for some parameters.
 
 A stochastic matrix is a matrix where each column or row is a
 unit simplex, meaning that each column (row) vector has non-negative
-values that sum to 1. The following example is a \(3 \times 4\)
+values that sum to 1. The following example is a $3 \times 4$
 column-stochastic matrix.
 
 $$
@@ -689,7 +689,7 @@ $$
 \end{bmatrix}
 $$
 
-An example of a \(3 \times 4\) row-stochastic matrix is the following.
+An example of a $3 \times 4$ row-stochastic matrix is the following.
 
 $$
 \begin{bmatrix}
@@ -731,7 +731,7 @@ As with simplexes, `column_stochastic_matrix` and `row_stochastic_matrix`
 variables are subject to validation, ensuring that each column (row)
 satisfies the simplex constraints. This validation accounts for
 floating-point imprecision, with checks performed up to a statically
-specified accuracy threshold \(\epsilon\).
+specified accuracy threshold $\epsilon$.
 
 #### Stability Considerations {-}