\documentclass{minimal}
\usepackage{amsmath}
\usepackage{hyperref}

\renewcommand{\emph}{\textbf}
\renewcommand\emph{\textbf}

\begin{document}

$a_1$
$a_+$
$a_-$
$a_=$
$a_{i + 1}$
$a_{i+1}$
$a_{\beta k}$
$a_{\hat k}$
$a_{\delta}$
$a_\delta$
$a_\rho$

$a^1$
$a^+$
$a^-$
$a^=$
$a^{i + 1}$
$a^{i+1}$
$a^{\hat k}$

\begin{equation}
  \bigl\lvert x \bigr\rvert
  \operatorname{test}(a) = 1
  \int f = \iiint g
\end{equation}

$\langle \| \\ \rangle$

% Concealed and styled (bold; italic)
\textup{asd $\alpha$}
\textbf{test}
\textit{test}
\textsl{test}
$4 \mathit{x} = \mathbf{test}$
$\mathrm{Re} = \cdots$

\begin{equation}
  \mathbb{P}(x \,|\, y) \!= \dots
  \label{eq:test-ref}
\end{equation}

Let's refer to \eqref{eq:test-ref}.

Let's refer to Eq. \ref{eq:test-ref}.

$\bar a$
$\bar{a}$

\begin{equation}
  \text{test}
  \textnormal{test $asd$}
  \textrm{test}
  \textup{test}
  \texttt{test}
  \textsf{test}
  \textsc{test}
\end{equation}

\href{url}{text}

\keyw{Piecewise \textbf{test} splines}

\begin{equation}
  \sum_{i=1}^n a_i = 0
\end{equation}

$$\lVert \mathrm{w} \rVert_2^2$$

\chapter{test}

\section{test}

\subsection{testing}

\subsubsection{testing}

Test tilde: \tilde n

Hello
\thinspace
\medspace
\thickspace
\bigskip
\hspace{1em}
\hfill
\vspace{1em}
\vfill
Moon

1 \, \: \; \! \quad \qquad 2

$1 \, \: \; \! \quad \qquad 2$

$Hello
\thinspace
\medspace
\thickspace
\bigskip
\hspace{1em}
\hfill
\vspace{1em}
Moon$

\end{document}