Am 24.10.2010 22:05, schrieb Venable:

First, I would like to include a boldface arrow. LyX allows me to
select Edit - Math - Text Style - Math Bold Series, and I can see in
the LaTeX source window that the markup is being applied (i.e.
\mathbf{\downarrow}), but when I export to PDF the arrow is not
discernibly bold.

You have to use \boldsymbol as described in sec. 11.2 "Bold Formulas" of the Math manual that you find in LyX's Help menu.

Second, I would like to know if there is a way to produce a negated
diagonal arrow that uses a scalable font. There is a \nsearrow symbol
in the mnsymbol package, but this appears to be a bitmap font and
therefore quite ugly.

You can negate everything using \not. So in your case type:
\not\searrow

(described in sec. 3.6 "Negations" of the Math manual)

regards Uwe

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