David Hewitt schrieb:

It's a 2-sided document with "headings" Page Style.

(1) The first problem is that the headings for sections/"high-level"
environments that start on odd-numbered pages show up on empty even-numbered
pages that follow. I'd like them not to appear on those empty pages, and if
possible to appear on the odd-numbered page where the section begins.

Both is possible I think. I don't know how exactly but you should find answers in the koma-script manual or its community: http://www.komascript.de The community page is German, but if nothing helps I would post there in English anyway - I'm sure you'll get an answer.

(2) The second problem I'm having is that the Figures in chapters don't
number like the equations. The equations happily begin with the Chapter
number plus ".[equation_number]", but the Figures in each chapter are just
numbered 1, 2, 3, ... Can I make the Figure numbers the same as the equation
numbering?

How this can be changed is described in section 3.3 of LyX's EmbeddedObjects 
manual.

regards Uwe

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