We say an edge e is "necessary" for the Minimum Spanning Tree T(G, E) if when we delete it, the weight of the MST increases. So we have that W(T(V, E \ {e})) > W(T(V, E))). Show that an edge e is necessary iff for every cycle C in G that contains it, e is not the maximum weight edge in this cycle (so there exists e 0 ∈ C such that we 0 > we)

Heat is added to a piston-cylinder device filled with 2 kg of air to raise its temperature 400 c from an initial temperature of t1 27 cand pressure of pi 1 mpa. the process is isobaric process. find a)-the final pressure p2 b)-the heat transfer to the air.

Apipe with an outside diameter of 15 cm is exposed to an ambient air and surrounding temperature of -20°c. the pipe has an outer surface temperature of 65°c and an emissivity of 0.85. if the rate of heat loss from the pipe surface is 0.95 kw per meter of length, the external convective heat transfer coefficient (h) is: (a) 12.5 w/m"k (b) 18.6 w/mk (c) 23.7 w/mk (d) 27.9 w/mk (e) 33.5 w/mk

Manometers are good examples of measuring instruments, nowadays they are not as common as before. a)-capacitive probe gauges b)-gravitational gauges deformation ) gauges d)-digital gauges