New PDF release: A treatise on plane and spherical trigonometry

By William Chauvenet

ISBN-10: 1406968250

ISBN-13: 9781406968255

This quantity is made out of electronic photographs created in the course of the collage of Michigan collage Library's renovation reformatting application.

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Extra info for A treatise on plane and spherical trigonometry

Example text

For the upper bound, T (n) = O(n2 ), we use the inductive hypothesis that T (n) ≤ cn 2 for some constant c > 0. By a similar derivation, we get that Solutions for Chapter 4: Recurrences 4-11 T (n) ≤ cn 2 if −2cn + n + c ≤ 0 or, equivalently, n(1 − 2c) + c ≤ 0. This condition holds for c = 1 and n ≥ 1. Thus, T (n) = (n 2 ) and T (n) = O(n 2 ), so we conclude that T (n) = √ h. T (n) = T ( n) + 1 (n2 ). The easy way to do this is with a change of variables, as on page 66 of the text. Let m = lg n and S(m) = T (2m ).

Meaning that whenever you interview a candidate who is better than your current ofÞce assistant, you must Þre the current ofÞce assistant and hire the candidate. Since you must have someone hired at all times, you will always hire the Þrst candidate that you interview. Goal: Determine what the price of this strategy will be. 5-2 Lecture Notes for Chapter 5: Probabilistic Analysis and Randomized Algorithms Pseudocode to model this scenario: Assumes that the candidates are numbered 1 to n and that after interviewing each candidate, we can determine if it’s better than the current ofÞce assistant.

Here, we have a recurrence with an exact function, rather than asymptotic notation, and the solution is also exact rather than asymptotic. ] 2. Induction: Basis: n = 1 ⇒ n lg n + n = 1 = T (n) Inductive step: Inductive hypothesis is that T (k) = k lg k + k for all k < n. We’ll use this inductive hypothesis for T (n/2). n +n T (n) = 2T 2 n n n lg + +n (by inductive hypothesis) = 2 2 2 2 n = n lg + n + n 2 = n(lg n − lg 2) + n + n = n lg n − n + n + n = n lg n + n . Lecture Notes for Chapter 4: Recurrences 4-3 Generally, we use asymptotic notation: • • • • We would write T (n) = 2T (n/2) + (n).

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A treatise on plane and spherical trigonometry by William Chauvenet

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