Let f ( x ) be a function from binary strings (of a fixed length
N ) to
binary strings. For the purposes of this problem, let’s say that f
( x )
has the equal difference property if the following is
satisfied?
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Let f ( x ) be a function from binary strings (of a fixed length N...
4. Let A be m n and B be m x 1 . Define f : IR"-> R by (a) Quote a previous problem to show that f has a minimum. Say that the minimum (b) Find Df. (Hint: Chain Rule using the function N from Problem 67.) occurs at y E R". (Note: it may be that A. y might be inconsistent.) B, since the equation A. X B (c) Apply the Interior Extreme Theorem to get an equation...
Let an be the number of strings of length n in 0, 1,2,3 which do not have a 000 substring. Find a recursion satisfied by the an. 7.
PROJECT 2-COUNTING SUBSETS (BINARY STRINGS Choose 6 letters of the English alphabet including all the different characters in your family name (If you have more than & diffecent characters, choose the first 61. Let X be the set di all lower case vensions of the letters you have chosen. Let S be the set of all binary strings of length 6 (0 Using cofrect set notation, list the elements in set X. (u) ust all the subsets of X with...
=(V, En) 5. Let n1 be an integer and define the graph Gn as follows {0,1}", the set of all binary strings of length n. Vn = Two vertices x and y are connected by an edge emu if and only if x and y differs in exactly one position. (a) (4 points) Draw the graph Gn for n = 1,2,3 (b) (4 points) For a general n 2 1, find |Vn and |En (c) (10 points) Prove that for...
(1) Let f : [n] [n] be a permutation. A fixed point of f is an element x e [n] such that f(x) - x. Now consider random permutations of [n] and let X be the random variable which represents the number of fixed points of a given permutation. (a) What is the probability that X 0? (b) What is the probability that X-n -2? (c) What is the probability that X-n-1? (d) What is the expectation of X? (Hint:...
Let f [n]n] be a permutation. A fixed point of f is an element x e [n] such that f(x)-x. Now consider random permutations of [n] and let X be the random variable which represents the number of fixed points of a given permutation. (a) What is the probability that X 0? (b) What is the probability that X 2? (c) What is the probability that X--1? (d) What is the expectation of X? (Hint: As usual, express X as...
Imprecise Counting - Long Runs in Binary Strings Let n=2^k for some positive integer k and consider the set Sn of all n-bit binary strings. Let c be an integer in {0,…,n−k}. Consider any j∈{1,…,n−k−c+1}. How many strings b1,…,bn∈Sn have bj,bj+1,…,bj+k+c−1=00…0? In other words, how many strings in Sn have k+c consecutive zeros beginning at position j? For each j∈{1,…,n−k+c+1}, let Xj be the subset of Sn consisting only of the strings counted in the previous question. Show that (n−k−c+1)∑(j=1)...
Let A be n × n with AT-A. (The matrix A is syrnmetric.) Let B be 1 × n and let c E R. Define f : Rn → R by f(x) = 2.7, A . x + B . x + c. Show that The function f is a quadratic function Let A be n × n with AT-A. (The matrix A is syrnmetric.) Let B be 1 × n and let c E R. Define f : Rn...
Let T be a binary tree with n nodes and let f() be the level numbering function of the positions of T f suggests a epteseniatñion of a binary tree Tty in el marabering function f suggests a f an aray-ased wucture A. with the lt of the array We show an etample of an an el ermbering funcion f sugests a represeuani sl Wr show an example of an antay baed rerjesctanisa od a A with the clement an...
Write a value-returning C++ function returns the average length of an array of strings. Name the function stringsAverageLength and use the following header: double stringsAverageLength(string array [], int n) { } where the parameter 'array' has 'n' strings and the return value is the average length of all of the strings in the array. For example, if the function is called like this: string cars[3] = { "Toyota", "Ford", "Tesla" }; cout << fixed << setprecision(2) << stringsAverageLength(cars, 3) <<...