Consider the function f(x) = α1x 2 1 + α2x 2 2 + · · · + αnx 2 n , where α1, αn are some constants. a) find all possible vectors [α1 α2 · · · αn] t for which f(x) is coercive. b) consider a vector [α1 α2 · · · αn] t for which f(x) is coercive. let us use the gradient algorithm with some initial point to minimize f(x). describe the convergence rate of the algorithm in terms of α1, α2, αn. c) consider a vector [α1 α2 · · · αn] t for which f(x) is coercive. find a matrix a such that when the problem is reformulated using the change of variables y = ax, the gradient algorithm has the fastest convergence rate.

Amass of m 1.5 kg of steam is contained in a closed rigid container. initially the pressure and temperature of the steam are: p 1.5 mpa and t 240°c (superheated state), respectively. then the temperature drops to t2= 100°c as the result of heat transfer to the surroundings. determine: a) quality of the steam at the end of the process, b) heat transfer with the surroundings. for: p1.5 mpa and t 240°c: enthalpy of superheated vapour is 2900 kj/kg, specific volume of superheated vapour is 0. 1483 m/kg, while for t 100°c: enthalpy of saturated liquid water is 419kj/kg, specific volume of saturated liquid water is 0.001043m/kg, enthalpy of saturated vapour is 2676 kj/kg, specific volume of saturated vapour is 1.672 m/kg and pressure is 0.1 mpa.

Afluid flows with a velocity field given by v=(x/t)i.. determine the local and convective accelerations when x=3 and t=1.

Aplate clutch has a single pair of mating friction surfaces 250-mm od by 175-mm id. the mean value of the coefficient of friction is 0.30, and the actuating force is 4 kn. a) find the maximum pressure and the torque capacity using the uniform-wear model. b) find the maximum pressure and the torque capacity using the uniform-pressure model.