Prove the following useful properties of dirac δ-functions (a) δ(ax) = (z) (b) zfic) =0 (c) f(x)5(-a) f(a)5(r d) δ(z-a (aメ0) a) ( dz9(x-a) ) where θ(x) is the step function defined as 1 if r 0 0 if r < 0 θ(z)

Because air contracts as it cools, the air pressure inside a freezer is typically lower than on the outside. why do ice cubes inside a freezer tend to shrink over time? the ice sublimes, and then the water vapor deposits as ice crystals on the sides of the freezer. the ice melts, and then the liquid freezes as ice crystals on the bottom of the freezer. the ice dissolves oxygen from the air, forming a denser crystalline matrix. the ice reacts chemically with carbon dioxide in the air, forming gaseous carbon compounds.