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Now examine the two terms in (5.71). By assumption, since L (p ) is the optimal length for p , we have L(p ) L (p ) 0. Similarly, the length of the extension of the optimal code for p has to have an average length at least as large as the optimal code for p [i.e., L(p) L (p) 0]. But the sum of two nonnegative terms can only be 0 if both of them are 0, which implies that L(p) = L (p) (i.e., the extension of the optimal code for p is optimal for p). Consequently, if we start with an optimal code for p with m 1 symbols and construct a code for m symbols by extending the codeword corresponding to pm 1 + pm , the new code is also optimal. Starting with a code for two elements, in which case the optimal code is obvious, we can by induction extend this result to prove the following theorem. Theorem 5.8.1 Huffman coding is optimal; that is, if C is a Huffman code and C is any other uniquely decodable code, L(C ) L(C ). Although we have proved the theorem for a binary alphabet, the proof can be extended to establishing optimality of the Huffman coding algorithm for a D-ary alphabet as well. Incidentally, we should remark that Huffman coding is a greedy algorithm in that it coalesces the two least likely symbols at each stage. The proof above shows that this local optimality ensures global optimality of the nal code. 5.9 SHANNON FANO ELIAS CODING

Werch, C. E., Moore, M., DiClemente, C. C., Owen, D. M., Jobli, E., & Bledsoe, R. (2003). A sport-based intervention for preventing alcohol use and promoting physical activity among adolescents. Journal of School Health, 73, 380 388. Whiteside, S. P., & Lynam, D. R. (2001). The five factor model and impulsivity: Using a structural model of personality to understand impulsivity. Personality and Individual Dif ferences, 30, 669 689. Wichstrom, L., & Pedersen, W. (2001). Use of anabolic-androgenic steroids in adolescence: Winning, looking good or being bad Journal of Studies on Alcohol, 62, 5 14. Wilson, G. S., Pritchard, M. E., & Schaffer, J. (2004). Athletic status and drinking behavior in college students: The influence of gender and coping styles. Journal of American College Health, 52, 269 273. Wolff, C. (1985, November 12). Alcohol level cited in Lindbergh crash. New York Times, p. B7. Wulf, S. (1988, October 3). Dropping weights. Sports Illustrated, 69, 18. Yesalis, C. E., Kennedy, N. J., Kopstein, A. N., & Bahrke, M. S. (1993). Anabolic-androgenic steroid use in the United States. Journal of the American Medical Association, 270, 1217 1221.

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to note that there are regions of varying stability on the full K hler curve. To the left of the o critical point, the droplets are in a stable equilibrium with the environment. That is to say, if a small amount of water molecules were added to the drop, its equilibrium pressure would be larger than the xed ambient value. Thus, the drop would evaporate, eventually returning to its equilibrium state. Similarly, if a small amount was evaporated from the droplet, its equilibrium pressure would be lower than the ambient and water would condense onto the drop until returning to its equilibrium state. To the right of the critical point, the droplet is in an unstable equilibrium. If it grew slightly, the equilibrium pressure would be lower than the ambient. Thus, water would condense onto the droplet. However, the more the droplet grows, the larger this difference becomes and the droplet would grow hindered only by the supply of water vapour. Similarly, if the droplet shrank a small amount, its equilibrium vapour pressure would be higher than the ambient and water would start to evaporate from the droplet. However, as more water evaporated, this difference would increase and water would continue to evaporate

s front of ce manager of The Times Hotel, you want to project the number of additional rooms you will need to book to achieve 100 percent occupancy for the night of April 15. Use the following historical data to determine the number of additional room reservations needed to achieve 100 percent occupancy: 500 rooms available, 100 con rmed reservations with a 5 percent no-show history, 200 guaranteed rooms with a 2 percent no-show history, 15 stayovers, 10 understays, and 45 walk-ins.

on the network seamlessly. In Wi-Fi there are analogous events of connecting and disconnecting from an access point, but there are two complications: A Wi-Fi device can get turned off without properly disconnecting. A Wi-Fi device can roam to a new access point. Both situations happen regularly. The former is dealt with by regular keep-alive signals. These signals approximate the electrical cable detection of Ethernet. The access points deal with the latter transparently. When a Wi-Fi node detects an access point on the same network but with a stronger signal, it sends a request to that access point to switch to it, and the access points communicate so that the old one gives up ownership and the new one takes it over. This takes a little time so if you re unlucky enough to be in a situation where your computer thinks two access points are about the same strength, you may bounce back and forth between them.

It is essential that the departmental contributory income statement provide maximum detail by showing each revenue and expense account to provide the information needed by management to conduct an effective and efficient evaluation.

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