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(a) Write the problem of nding the maximum ow from S to T as a linear program (b) Write down the dual of this linear program There should be a dual variable for each edge of the network and for each vertex other than S, T Now we ll solve the same problem in full generality Recall the linear program for a general maximum ow problem (Section 72) (c) Write down the dual of this general ow LP, using a variable ye for each edge and xu for each vertex u = s, t (d) Show that any solution to the general dual LP must satisfy the following property: for any directed path from s to t in the network, the sum of the ye values along the path must be at least 1 (e) What are the intuitive meanings of the dual variables Show that any s t cut in the network can be translated into a dual feasible solution whose cost is exactly the capacity of that cut 726 In a satis able system of linear inequalities a11 x1 + + a1n xn am1 x1 + + amn xn b1 bm

y 6 ( 2,4) (3,5)

In radio receivers and transmitters, transformers can be categorized generally by the method of construction used Some have primary and secondary windings, just like utility and audio units Others employ transmission-line sections These are the two most common types of transformer found at radio frequencies

2 x 6 4 2 2 4 6 m= y/ x = (5 4) / [3 ( 2)] = 1/5 2 4 6

we describe the jth inequality as forced-equal if it is satis ed with equality by every solution x = (x1 , , xn ) of the system Equivalently, i aji xi bj is not forced-equal if there exists an x that satis es the whole system and such that i aji xi < bj For example, in x1 + x 2 x1 x2 x1 x2 2

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Switching the order We can switch the points (x1, y1) and (x2, y2) and still get the same slope when we calculate it as above That s because both the numerator and the denominator end up being additive inverses (exact negatives) of what they were before Let s take

Wire-Wound Types In wire-wound RF transformers, powdered-iron cores can be used up to quite high frequencies Toroidal cores are common, because they are self-shielding (all of the magnetic flux is confined within the core material) The number of turns depends on the frequency, and also on the permeability of the core In high-power applications, air-core coils are often preferred Although air has low permeability, it has negligible hysteresis loss, and will not heat up or fracture as powdered-iron cores sometimes do The disadvantage of air-core coils is that some of the magnetic flux extends outside of the coil This affects the performance of the transformer when it must be placed in a cramped space, such as in a transmitter final-amplifier compartment A major advantage of coil-type transformers, especially when they are wound on toroidal cores, is that they can be made to work over a wide band of frequencies, such as from 35 MHz to 30 MHz These are called broadband transformers Transmission-Line Types As you recall, any transmission line has a characteristic impedance, or Zo, that depends on the line construction This property is sometimes used to make impedance transformers out of coaxial or parallel-wire line Transmission-line transformers are always made from quarter-wave sections From the previous chapter, remember the formula for the length of a quarter-wave section:

the rst two inequalities are forced-equal, while the third and fourth are not A solution x to the system is called characteristic if, for every inequality I that is not forced-equal, x satis es I without equality In the instance above, such a solution is (x1 , x2 ) = ( 1, 3), for which x1 < 1 and x2 < 0 while x1 + x2 = 2 and x1 x2 = 2 (a) Show that any satis able system has a characteristic solution (b) Given a satis able system of linear inequalities, show how to use linear programming to determine which inequalities are forced-equal, and to nd a characteristic solution 727 Show that the change-making problem (Exercise 617) can be formulated as an integer linear program Can we solve this program as an LP, in the certainty that the solution will turn out to be integral (as in the case of bipartite matching) Either prove it or give a counterexample

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