*snicker*
Ok, so Alivia is a form of the name Olivia. And Olivia means "olive tree"...which, according the Shakespear, means bringer of peace...bearer of peace...to make peace...or to offer peace. So, what does that tell us about Alivia in the book?
Well, nothing really...but, this is starting to sound good.
Alivia helps Rand "die"...by helping make a peace between him and Lews Therin (and possibly Moridin since they seem to be connected also). So, by making peace, in this case, the two or three men form into one person...and their individuality dies.

*whew* My head hurts after that!

-Garrison Bain-