Adverbs

Adverbs
 

Why asidua y persistentemente and not asiduo y persistentemente?

it can be asidua or asiduo depending on the gender of the noun it monifies. Asidua/o is an adjective.

Actually, what you're dealing with is a compound adverb (two or more adverbs in a series). Both of the words you posted are adverbs, but the -mente ending is dropped for all but the last adverb when they are grouped together.
This is proper usage and is very common.

Here's an example that uses lentamente, claramente and deliberadamente in a series:
Él habló lenta, clara y deliberadamente.
= He spoke slowly, clearly and deliberately.

so generalmente is a compound adverb too. this is useful information

No. "generalmente" is just an adverb. As Rusty explained, when he said "compound adverbs", he is referring to two or more adverbs used in series, in which case only the last one gets the -mente suffix.

oh ok now I understand..