Suppose that the function f satisfies the recurrence rela- tion f (n)2f(Vn)+1 whenever n is a...
Suppose f:N → N satisfies the recurrence f(n+1) = f(n) 7. Note that this is not enough information to define the function, since we don't have an initial condition. For each of the initial conditions below, find the value of f(7). a. f(0) = 1. $(7) = b. f(0) = 5. f(7) = c. f(0) = 19. f(7) = d. f(0) = 249. f(7) =
5. Let F(n, m) denote the number of paths from top-left cell to bottom-right cell in a (n x m) grid (that only permits moving right or moving down). It satisfies the recurrence relation F(n, m) F(n-1, m) + F(n, m-1) What should be the initial condition for this recurrence relation? (Hint: What would be the number of paths if there was only a single row or a single column in the grid?)[5] Convince yourself that F(n, m) gives correct...
if possible solve part d in detail. a) fi(n) n2+ 45 n log n b) f:(n)-1o+ n3 +856 c) f3(n) 16 vn log n 2. Use the functions in part 1 a) Isfi(n) in O(f(n)), Ω(fg(n)), or Θ((6(n))? b) Isfi(n) in O(f(n)), Ω(f,(n)), or Θ((fs(n))? c) Ísf3(n) in O(f(n)), Ω(f(n)), or Θ(f(n))? d) Under what condition, if any, would the "less efficient" algorithm execute more quickly than the "more efficient" algorithm in question c? Explain Give explanations for your answers...
4. RWI 4.5.13). Suppose that the sequence (xn) satisfies n 1,2,.. X2ax.1 + bx and that 0< r<R satisfy - az- b (z-rz- R) (a) Show that x O(R" ). O(2") is false. (b) Give an example with r R 2 for which x, (c) What asymptotic (big-oh) estimate for (x) can you give in general if r R> 0?
(a) Suppose that f is continuous on [0, 1] and f(o) = f(1). Let n be 20. any natural number. Prove that there is some number x such that f fx+1/m), as shown in Figure 16 for n 4. Hint: Consider the function g(x) = f(x)-f(x + 1/n); what would be true if g(x)ヂ0 for all x? "(b) Suppose 0 < a 1, but that a is not equal to 1/n for any natural number n. Find a function f...
16: Problem 8 Previous Problem ListNext 1 point) 1) Suppose that f(x) is a function that is positive and decreasing. Recall that by the integral test: f(z) dz < Σ f(n) Recall that e-Σ. ,,Suppose that tor each positive integer k f(k)- Find an upper bound Bor f(z) dz 2) A function is given by ts values may be found in tables. Make the change of variables y In(4) to express 1-4 d as a constant C times h(3). Find...
An m×n array A of real numbers is a Monge array if for all i,j,k, and l such that 1≤i<k≤m and 1≤j<l≤n , we have >A[i,j]+a[k,l]≤A[i,l]+A[k,j]> In other words, whenever we pick two rows and two columns of a Monge array and consider the four elements at the intersections of the rows and columns, the sum of the upper-left and lower-right elements is less than or equal to the sum of the lower-left and upper-right elements. For example, the following...
F1. need help solving this problem. 1. (25 pts) Here's a neat theorem. Suppose that f la, b] [a, b] is continuous; then f will always map some s-value to itself (a so-called fixed point): i.e. 3 c E (a, b) for which f(c)-c (a) Give a "visual proof" of this theorem. Hint: take your inspiration from our "visual proofs" of Theorem 15 and IVT And notice here that the domain and range of f are the same interval; this...
Suppose firm j’s output is given by yj = n 1−α j , where 0 < α < 1 (α is a parameter). Suppose the firm must pay a fixed cost b < α if it wants to operate. That is, the firm’s profits are given by π (nj ) = 0 , if nj = 0 and π (nj ) = n 1−α j − wnj − b , if nj > 0 where w is the wage. (a)...
Consider the problem of estimating π using a unit square centered at (1/2, 1/2) and an inscribed circle inside the square. We will estimate π by simulating n darts. For the nth dart, if the dart is inside the circle, then we return In = 1; otherwise, we return In = 0. 1. Are I1, I2, · · · , In independent? Under what assumption? 2. Are I1, I2, · · · , In identically distributed? 3. Let p represent...