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) (Postcondition: The entire sequence s is sorted) 1 Do steps 2 3 for i = n/2 1 down to 0 2 Apply the heapify algorithm to the subsequence {si sn 1} 3 (Invariant: every root-to-leaf path in s is nonincreasing) 4 Do steps 5 7 for i = n 1 down to 1 5 Swap si with s0 6 (Invariant: The subsequence {si sn 1} is sorted) 7 Apply the heapify algorithm to the subsequence {s0 si 1} Algorithm 149 The Heapify (Preconditions: ss = {si sj 1} is a subsequence of j i ordinal values, and both subsequences {si+1 sj 1} and {si+2 sj 1} have the heap property) (Postcondition: ss itself has the heap property) 1 Let t = s2i+1 2 Let sk = max{s2i+1, s2i+2}, so k = 2i+1 or 2i+2, the index of the larger child 3.

CHAP. 1]

If t < sk , do steps 4 6 4 Set si = sk 5 Set i = k 6 If i < n/2 and si < max{s2i+1, s2i+2}, repeat steps 1 4 7 Set sk = t There are two aspects to these algorithms that distinguish them from the methods outlined in 12 The heaps here are in the reverse order, so each root-to-leaf path is descending And these algorithms use 0-based indexing The reverse order guarantees that heapify will always leave the largest element at the root of the subsequence Using 0-base indexing instead of 1based indexing renders the sort() method consistent with all the other sort() methods at the expense of making the code a little more complicated EXAMPLE 1410 The Heap Sort.

For the two-port network of Fig. 1-20, (a) nd the voltage-gain ratio V2 =V1 in terms of the z parameters, and then (b) evaluate the ratio, using the h-parameter values given in Problem 1.15 and the results of Problem 1.36. Ans: a z21 RL = z11 RL z11 z22 z12 z21 ; b 200

If someone tried to patent a three-wheeled suitcase, arguing that only two- and four-wheeled suitcases existed prior to the patent application, the patent would probably be denied as being obvious A three-wheeled suitcase might be novel, and it might be useful, but once someone comes up with the idea of a wheeled suitcase, the number of wheels seems to be a minor detail something obvious to someone in the business of designing suitcases Since 1981, software has been patentable in the United States, if the software is part of a patentable device In general, software is not patentable because scientific truths and the mathematical expressions of scientific truths are not patentable The 1981 case involved a patent for a rubber curing device which incorporated a computer for control Since software was part of a patentable rubber molding device, the software was patentable.

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Find the current-gain ratio I2 =I1 for the two-port network of Fig. 1-20 in terms of the h parameters. Ans: h21 = 1 h22 RL

Find the current-gain ratio I2 =I1 for the two-port network of Fig. 1-20 in terms of the z parameters. Ans: z21 = z22 RL

public static void sort(int[] a) { // POSTCONDITION: a[0] <= a[1] <= ... <= a[a.length-1]; int n = a.length; for (int i = n/2 - 1; i >= 0; i--) { // step 1 heapify(a, i, n); // step 2 }

Mathematical algorithms remain unpatentable, so a new sorting algorithm could not be protected by a patent (It could be protected as a trade secret, of course) Since 1981, software has been patentable if the software is part of an invention of a new machine or process A key to making the distinction has been whether the software manipulates measurements obtained from real-world sensors A program controlling a toaster, or a robotic warehouse, for example, would likely be patentable Patents granted to software have not been as protective as patents granted to more traditional inventions When deciding whether an invention is novel, patent examiners usually rely on inspection of earlier patents Since software has only recently become patentable, the prior art of the field is difficult for examiners from outside the field to learn.

Determine the Thevenin equivalent voltage and impedance, in terms of the z parameters, looking right into port 1 of the two-port network of Fig. 1-20 if RL is replaced with an independent dc voltage source Vd , connected such that V2 Vd . Ans: VTh z12 Vd =z22 ; ZTh z11 z22 z12 z21 =z22

CHAP. 14]

Find the Thevenin equivalent voltage and impedance, in terms of the h parameters, looking right into port 1 of the network of Fig. 1-20 if RL is replaced with a voltage-controlled current source such that I2 V1 , where > 0 and the h parameters are understood to be positive. Ans: VTh 0; ZTh h11 h22 h12 h21 = h22 h12

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Determine the driving-point impedance (the input impedance with all independent sources deactivated) of the two-port network of Fig. 1-20. Ans: z11 RL z11 z22 z12 z21 = z22 RL

As a result, many bad patents have been issued for software, with the result that software patents are more subject to challenge than other types of patents, and the individual holding a software.