What kind of bicycle clamps are these? I'm hoping to find a specific name because the bolt for them snapped and I can't find it, along with this roundish cylinder thing that used to hold it together; I found a screw that doesn't quite get it snug enough, but it holds it together to take pictures.
Thats just a standard single-bolt seatpost clamp -- almost every seatpost on the market uses that clamping mechanism. Laprade was the original company who designed that type, but nobody uses that name with it.
In order to get the (clamp) nut/bar (cylinder) and bolt, you can check your local bike shop to see if they have something that fits in their spare parts bin or contact the manufacturer of the seatpost to get one or canibalize a donor seatpost.