Cherokee,
As you've may have already figured out, it's almost impossible to get rid of them. If you cut them, burn them, mow them, it doesn't matter, they come back stronger. The only natural way is over tilling. Till a half a dozen times over a course of a few weeks. If you're not concerned with doing it the natural way, then a herbicide like Glyphosate (Roundup) and triclopyr (Brush-Be-Gone) are good choices. The issue is, they have to be sprayed while they are active. When they are active they are fruiting and it could cause a problem if someone eats the sprayed fruit. If you're sure no one is going to eat the fruit, or if you want to put up a sign, it's up to you. If you spray when it starts losing it's leaves and going dormant, it will be useless. Also, be sure to follow the directions on the herbicide closely.
Let me know if I can be of further assistance
Chase
Handywoman
Landscaping, Gardening, Pests, Interior/Exterior Decorating, Feng Shui, Pools/Ponds