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A concept that fascinates Bollinger is what he calls the deviation from the expected When you build a trading system, he says, you have an expectation that you will have at least 60 percent winners, for example And the winners will be about 15 times the size of the losers, and you ll rarely get a string of winners or losers greater than N or M (N is the expectation of the average streak of winning trades, and M is the expectation of the length of the average losing streak) He suggests that you chart out each of these variables as your system goes along And if any of them start to change, whether it s for better or for worse, you know that the character of the market is changing and that you may be in trouble The problem, Bollinger says, is that people ignore changes to their system if it s for the better Why Because you re making money, he explains Let s say you steadily have 60 percent winners Then suddenly over the next three months you have 65 or 70 percent winners That s a deviation from the expected That s telling you there s been a change, even though it s in your favor It s as important as your going from 60 percent down to 50 percent But people ignore the change because it benefits them They only pay attention to the negative change, but each contains a warning that there has been a change in the underlying market

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The right-hand law works with division If you have a sum or difference as the dividend and the single number as the divisor, you can use the distributive law for division over addition or division over subtraction If a and b are any integers, and if c is any nonzero integer, then

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212 How many lines, as a function of n (in ( ) form), does the following program print Write a recurrence and solve it You may assume n is a power of 2 function f(n) if n > 1: print_line( still going ) f(n/2) f(n/2) 213 A binary tree is full if all of its vertices have either zero or two children Let B n denote the number of full binary trees with n vertices (a) By drawing out all full binary trees with 3, 5, or 7 vertices, determine the exact values of B3 , B5 , and B7 Why have we left out even numbers of vertices, like B4 (b) For general n, derive a recurrence relation for Bn (c) Show by induction that Bn is (2n ) 214 You are given an array of n elements, and you notice that some of the elements are duplicates; that is, they appear more than once in the array Show how to remove all duplicates from the array in time O(n log n) 215 In our median- nding algorithm (Section 24), a basic primitive is the split operation, which takes as input an array S and a value v and then divides S into three sets: the elements less than v, the elements equal to v, and the elements greater than v Show how to implement this split operation in place, that is, without allocating new memory 216 You are given an in nite array A[ ] in which the rst n cells contain integers in sorted order and the rest of the cells are lled with You are not given the value of n Describe an algorithm that takes an integer x as input and nds a position in the array containing x, if such a position exists, in O(log n) time (If you are disturbed by the fact that the array A has in nite length, assume instead that it is of length n, but that you don t know this length, and that the implementation of the array data type in your programming language returns the error message whenever elements A[i] with i > n are accessed) 217 Given a sorted array of distinct integers A[1, , n], you want to nd out whether there is an index i for which A[i] = i Give a divide-and-conquer algorithm that runs in time O(log n) 218 Consider the task of searching a sorted array A[1 n] for a given element x: a task we usually perform by binary search in time O(log n) Show that any algorithm that accesses the array only via comparisons (that is, by asking questions of the form is A[i] z ), must take (log n) steps.

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