You are given a graph g = (v, e) with positive edge weights, and a minimum spanning tree t = (v , e ′ ) with respect to these weights; you may assume g and t are given as adjacency lists. now suppose the weight of a particular edge e ∈ e is modified from w(e) to a new value wˆ(e). you wish to quickly update the minimum spanning tree t to reflect this change, without recomputing the entire tree from scratch. there are four cases. in each case give a linear-time algorithm for updating the tree.

Air is to be cooled in the evaporator section of a refrigerator by passing it over a bank of 0.8-cm-outer-diameter and 0.4-m-long tubes inside which the refrigerant is evaporating at -20°c. air approaches the tube bank in the normal direction at 0°c and 1 atm with a mean velocity of 4 m/s. the tubes are arranged in-line with longitudinal and transverse pitches of sl- st 1.5 cm. there are 30 rows in the flow direction with 15 tubes in each row. determine (a) the refrigeration capacity of this system and (b) pressure drop across the tube bank. evaluate the air properties at an assumed mean temperature of -5°c and 1 atm. is this a good assumption?

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.

Burgers vector is generally parallel to the dislocation line. a)-true b)-false