Mathematics for Computer Science
12.8. Another Characterization for Planar Graphs 493 (a) (b) (c) (d) (e) (f) e 1 v 1 v 2 e 2 v 3 Figure 12.16 One method by whic ...
Chapter 12 Planar Graphs494 s t u r v x y w Problems for Section 12.8 Exam Problems Problem 12.2. b c d e a g h i f j k l m n o ...
12.8. Another Characterization for Planar Graphs 495 Problem 12.3. (a)Give an example of a planar graph with two planar embeddin ...
Chapter 12 Planar Graphs496 (d)How many faces does a planar graph from part c have? (e)How many distinct isomorphisms are there ...
12.8. Another Characterization for Planar Graphs 497 a b d c e e figure 1 figure 2 figure 3 figure 4 a b d c a b d c a b d c Fig ...
Chapter 12 Planar Graphs498 Problem 12.9. (a)Prove Lemma(Switch Edges).Suppose that, starting from some embeddings of planar gra ...
III Counting ...
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Introduction Counting seems easy enough: 1, 2, 3, 4, etc. This direct approach works well for counting simple things—like your t ...
Part III Counting502 Chapter 14 describes the most basic rules for determining the cardinality of a set. These rules are actuall ...
13 Sums and Asymptotics Sums and products arise regularly in the analysis of algorithms, financial appli- cations, physical prob ...
Chapter 13 Sums and Asymptotics504 chapter we will use this approach to find a good closed-form approximation to the factorial f ...
13.1. The Value of an Annuity 505 Here is why the interest ratepmatters. Ten dollars invested today at interest rate pwill becom ...
Chapter 13 Sums and Asymptotics506 The result of the subtraction is SxSD 1 xnC^1 : Solving forSgives the desired closed-form exp ...
13.1. The Value of an Annuity 507 Proof. X^1 iD 0 xiWWDnlim!1 Xn iD 0 xi D lim n!1 1 xnC^1 1 x (by equation 13.2) D 1 1 x : The ...
Chapter 13 Sums and Asymptotics508 tion 13.2 and Theorem 13.1.1: 1 C1=2C1=4CD X^1 iD 0 1 2 i D 1 1 .1=2/ D 2 (13.6) 0:9999 ...
13.1. The Value of an Annuity 509 differentiating equation 13.2 leads to: d dx (^) n 1 X iD 0 xi ! D d dx 1 xn 1 x : (13.11) ...
Chapter 13 Sums and Asymptotics510 As a consequence, suppose that there is an annuity that paysimdollars at the end of each year ...
13.2. Sums of Powers 511 yours—to show you how Gauss is supposed to have proved equation 13.1 when he was a young boy. Gauss’s i ...
Chapter 13 Sums and Asymptotics512 The point is that if the desired formula turns out to be a polynomial, then once you get an e ...
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