Alignment tests for in-line mathematics

Alignment to context

Juxtaposing normal and math characters.

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T$T$h$h$e$e$ q$q$u$u$i$i$c$c$k$k$ b$b$r$r$o$o$w$w$n$n$ f$f$o$o$x$x$ j$j$u$u$m$m$p$p$s$s$ o$o$v$v$e$e$r$r$ t$t$h$h$e$e$ l$l$a$a$z$z$y$y$ d$d$o$o$g$g$.
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T$\mathbf{T}$h$\mathbf{h}$e$\mathbf{e}$ q$\mathbf{q}$u$\mathbf{u}$i$\mathbf{i}$c$\mathbf{c}$k$\mathbf{k}$ b$\mathbf{b}$r$\mathbf{r}$o$\mathbf{o}$w$\mathbf{w}$n$\mathbf{n}$ f$\mathbf{f}$o$\mathbf{o}$x$\mathbf{x}$ j$\mathbf{j}$u$\mathbf{u}$m$\mathbf{m}$p$\mathbf{p}$s$\mathbf{s}$ o$\mathbf{o}$v$\mathbf{v}$e$\mathbf{e}$r$\mathbf{r}$ t$\mathbf{t}$h$\mathbf{h}$e$\mathbf{e}$ l$\mathbf{l}$a$\mathbf{a}$z$\mathbf{z}$y$\mathbf{y}$ d$\mathbf{d}$o$\mathbf{o}$g$\mathbf{g}$.
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Baseline detachment

Showing characters that rise completely above the baseline, or fall completely below it.

Unified expressions (top) vs. disjoint, separately-delimited characters (bottom).

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$x^2 + y^2 = z^2$
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$x$$^2$ $+$ $y$$^2$ $=$ $z$$^2$
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$\mathscr{T}_d \to \mathscr{T}_r$
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$\mathscr{T}$$_d$ $\to$ $\mathscr{T}$$_r$
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$x\prime - y\prime$
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$x$$\prime$ $-$ $y$$\prime$
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$\underline{}\underline{x}\underline{}x\underline{}\underline{x}\underline{}$
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$\underline{}$$\underline{x}$$\underline{}$$x$$\underline{}$$\underline{x}$$\underline{}$
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Complex expressions placed in-line

Using in-line delimiters (top) vs. block delimiters (bottom).

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Consider $\sum_{t=0}^\infty p\epsilon^t \prod_{u=0}^{t-1} (1 - p\epsilon^u)$ for instance.
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Consider $$\sum_{t=0}^\infty p\epsilon^t \prod_{u=0}^{t-1} (1 - p\epsilon^u)$$ for instance.
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