For each of the following questions, briefly explain your answer. (20 points) a. if i prove an algorithm takes 2 ) worst-case time, is it possible that it takes o(n) on some inputs? b. if i prove an algorithm takes 2 ) worst-case time, is it possible that it takes o(n) on all inputs? c. if i prove an algorithm takes θ 2 ) worst-case time, is it possible that it takes o(n) on some inputs? d. if i prove an algorithm takes θ 2 ) worst-case time, is it possible that it takes o(n) on all inputs?

If the temperature were raised very high, classically what would we expect the heat capacity per object to be for this one-dimensional system? give a numerical value. chigh t = __ j/k/object (one reason for the discrepancy is that the high-temperature limit assumes that the number of oscillators is large (n > > 1), which is not the case in this tiny system.)

Complete the statement about memory improvement. ( )involves organizing and breaking down information into easier groups to expand capacity. rehearsal is the verbal repetition of information. these techniques are especially important for the improvement of ( ) memory.

The ultracentrifuge is an important tool for separating and analyzing proteins. because of the enormous centripetal accelerations, the centrifuge must be carefully balanced, with each sample matched by a sample of identical mass on the opposite side. any difference in the masses of opposing samples creates a net force on the shaft of the rotor, potentially leading to a catastrophic failure of the apparatus. suppose a scientist makes a slight error in sample preparation and one sample has a mass 10 mg larger than the opposing sample. if the samples are 12 cm from the axis of the rotor and the ultracentrifuge spins at 70,000 rpm, what is the magnitude of the net force on the rotor due to the unbalanced samples? ( be thorough on your answer)