Problem A: For any integer k, 0 s k sS n, determine the number of vectors...
Question 8: For any integer n 20 and any real number x with 0<<1, define the function (Using the ratio test from calculus, it can be shown that this infinite series converges for any fixed integer n.) Determine a closed form expression for Fo(x). (You may use any result that was proven in class.) Let n 21 be an integer and let r be a real number with 0<< 1. Prove that 'n-1(2), n where 1 denotes the derivative of...
Problem 2: Recall that pi(n) denotes the number of integer partitions of n with exactly k parts. Show that Pk(n)an- m11 n20 Problem 2: Recall that pi(n) denotes the number of integer partitions of n with exactly k parts. Show that Pk(n)an- m11 n20
Let k 21 be a positive integer, and let r R be a non-zero real number. For any real number e, we would like to show that for all 0 SjSk-, the function satisfies the advancement operator equation (A -r)f0 (a) Show that this is true whenever J-0. You can use the fact that f(n) = crn satisfies (A-r)f = 0. (b) Suppose fm n) satisfies the equation when m s k-2 for every choice of c. Show that )...
7. Consider the set of vectors that is, the set of vectors with n components, each of which is either 0 or 1, Let Ω0dd be the subset of S2n consisting of all vectors . . . xn] for which Σ-12i is odd. (a) How many vectors are in Ωη? How many vectors are in mid? (b) If n is even, let k-n -1; if n is odd let k- n. Explain why the sum 3 odd must be the...
7. Consider the set of vectors that is, the set of vectors with n components, each of which is either 0 or 1. Let 2odd be the subset of Ωη consisting of all vectors u= [r1 xn] for which Σ-1 xỉ 1s odd (a) How many vectors are in Ωη? How many vectors are in :dd? (b) If n is even. let k = n-1: if n is odd let k = n. Explain why the sum odd must be...
7. Consider the set of vectors that is, the set of vectors with n components, each of which is either 0 or 1. Let 2odd be the subset of Ωη consisting of all vectors u-Pi . . . xn] for which Ση:l xi is odd (a) How many vectors are in Ω? How many vectors are in Ω dd? (b) If n is even, let k n-1: if n is odd let k = n. Explain why the sum odd...
EXPLAIN STEP BY STEP In Exercises 13 through 18 determine if the set of vectors S forms a subspace of the given vector space. Give reasons why S either is or is not a subspace. xn) in 13. S is the set of vectors of the form (x1, X2, ..., xn) in R”, with the x; real numbers and x2 = x4. 14. S is the set of vectors of the form (x1, X2, . R”, with the xị real...
Question 6: Let n 2 1 be an integer and let A[1...n] be an array that stores a permutation of the set { 1, 2, . .. , n). If the array A s sorted. then Ak] = k for k = 1.2. .., n and, thus. TL k-1 If the array A is not sorted and Ak-i, where iメk, then Ak-서 is equal to the "distance" that the valuei must move in order to make the array sorted. Thus,...
Given a real number x and a positive integer k, determine the number of multiplications used to find x^(2^i) starting with x and successively squaring ( to find x^2, x^4 and so on). Is this a more efficient way to find x^(2^i) than by multiplying x by itself the appropriate number of times? Give the C++ code to work this problem, if possible with recursion, if not explain why?
For a given k, 1 ≤ k ≤ n, how many vectors (x1, x2, . . . , xk) are there for which each xi is a positive integer such that 1 ≤ x1 < x2 < · · · < xk ≤ n?