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In Windows, TexWorks loads the missing sty packages itself. In Linux it doesn't: how is this sty loading managed in Linux?

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Assuming you are using texlive, the common latex distribution these days, search path management is performed by kpsepath. To find out where latex searches for sty-files type kpsepath tex. By simply typing kpsepath you get a list of all the different search paths there are.

For simple modification of the search path you can set the environment-variable TEXINPUTS, e.g. export TEXINPUTS=${TEXINPUTS}:/path/to/my/sty/files

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That depends on how the Linux distribution handles LaTeX and its packages. Older Fedora used a huge texlive package, containing everything. It has recently been split up into lots of individual packages handled by the distribution's package machinery. Others might go for the native TeXlive way via tlmgr.

In Fedora 18, you can run yum install 'tex(thepackage.sty)' as root.

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