Max Flow/Min Cut Theorem applied to a Network Flow Diagram

Please check the attached file for the network diagram

A network of railway lines connects the main lines entering and leaving a city. Speed limits, track reconstruction, and train length restrictions lead to the flow diagram below, where the numbers represent how many cars can pass per hour. What is the maximal flow in cars per hour from the west to the east?

Is it true that if I multiply all the edge capacities in an s-t flow problem by a positive constant k > 0, then the maximum flow increases by the same factor of k.

The Kodiak Oil Company owns a pipeline network that is used to convey oil from its source (Node 1) to several storage locations. The networkflow is shown on the attachment. Due to varying pipe sizes, the flow capacities vary (shown below in รข??000s of gallons per hour). By selectively opening and closing sections of the pipeli

1. Find the maximum flow and the associated minimum capacity cut for the following network, by using flow augmenting path algorithm (in the order of "first labeled, first scannred"). *see attachment for diagram*
2. A maximum capacity flow augmenting path is an augmenting path such that we can increase the flow of the network

1) Let f, g be defined on R and let c in R. Suppose that lim f = b and that g is continuous at b. Show that lim g 0 f = g(b)
Note:
R: real numbers
g 0 f means composition of f and g
2) Let A = [0, 1) U (1,2]. Let B = [0, 1] U [2, 3]. Does the conclusion of the maximum-minimum theorem always hold for a function f: A

(d) Does the conclusion of the Maximum-Minimum Theorem always hold for a bounded function f : R --> R that is continuous on R? Prove or give a counterexample.
(a) Fix a, b E R, a < b. Prove that if f [a, b] -->R is continuous on [a, b] and f(x)≠0 for all x E [a, b], then 1/f(x) is bounded on [a, b].
(b) Find a, b E R, a

The diagram shows a gear pump. The maximum diameter of the gear wheels is 80 mm, the diameter of the roots of the teeth is 50 mm and the tooth width is 20mm. If the pump is driven at 500 rpm, please estimate the volume of oil pumped every minute.
If the pressure difference across the pump is 50 bar, what shaft power has to be

Let G = (V, E) be a flownetwork with source s, sink t, and suppose each edge e E has capacity c(e) = 1. Assume also, for convenience, that |E| = (V).
a. Suppose we implement the Ford-Fulkerson maximum-flow algorithm by using depth-first search to find augmenting paths in the residual graph. what is the wo