Use a computer to plot formula 2.22 directly, as follows. Define  z = qA/q, so that (l-z) = qB/q. Then, aside from an overall constant that we'll ignore, the multiplicity function is [4z(1-z)]N, where z ranges from 0 to 1 and the factor of 4 ensures that the height of the peak is equal to 1 for any N. Plot this function for N = 1, 10, 100, 1000, and 10,000. Observe how the width of the peak decreases as N increases.

The factorial function f(n) = n! is defined by n! = 1 2 3... n whenever n is a positive integer, and O! = 1. Provide big-O estimates for the factorial function and the logarithm of the factorial function. As an illustration, 1! = 1, 2! = 1 2=2, 3! = 1 23 = 6, and 4! = 1 234=24.Take note of how quickly the function n! expands. Consider the following: 20! = 2,432,902,008,176,640,000.

Please answer question.

1. Consider the following functions of n. (1) (2) (3) (4) (5) (6) (7) (8) (9) fi(n) = n, if n = 2.k for k = 0, 1, 2,... = n², otherwise. f2(n) = n, if n = 3.k for k= 0, 1, 2,... = n², otherwise. f3(n) = log n. Answer the following questions and prove your answ Yes or No. Yes or No. Yes or No. Yes or No. Yes or No. Yes or No. Yes or No. Yes or No. Yes or No. fi(n) = 2(f₂(n)), f2(n) = Q(fi(n)), fi(n) = 0 (f₂(n)), f₂(n) = 0 (fi(n)), fi(n) = O(n), f₂(n) = O(n²), f3(n) = O(√n), f3(n) = O(fi(n)), 22n = 0(2¹), swer for each one.

N.png ... If gcd(a, m) = 1, then the unique solution xo to ax = b (mod m) is equal to (1) xo = ab (mod m) (2) xo = ab (mod m), where b is the inverse of b. (3) xo = ām (mod m), where ā is the inverse of a. (4) xo = āb (mod m), where ā is the inverse of a. A Choice (2) (B) Choice (1) Choice (4) (D) Choice (3)

For the following pairs of functions, first decide whether f(n) dominates g(n), or g(n) dominates f(n); then decide whether f = O(g), or f = Ω(g), or f = Θ(g), and briefly explains. i. f(n)=n2,g(n)=1000n+30 √1 ii. f(n) = n, g(n) = n3 iii. f(n) = 10 log2 n, g(n) = log10(n3) iv. f(n)=n 100 n ,g(n)=1.2 . 4 nn v. f(n)=2 ,g(n)=2.01 .

Consider the formula rv p→qv¬r Fill in the blanks with names of the columns of a truth table for this formula. To insert special symbols, copy them from the given formula. If there is more than one way to label columns, put them in left to right order. Do not simplify the formula. For example, if the formula is (p →q) ^ - p, then insert the following in the five blanks (without numbers, numbers are just to denote which blank it is): 1) p 2) q 3) p q 4) -p 5) (p →q) ^ ¬p Note that the variable columns are labeled in alphabetical order (p, then q, then r...), p →q is listed before -p (left to right), and there are no outer parentheses on any formulas.

Let f (f(n) and g(n)) be asymptotically nonnegative functions. Using the basic definition of Θ notation, prove that max(f(n), g(n)) = Θ(f(n) + g(n)),

Consider the following truth table where A, B, C, and D are considered as inputs and F is the output: F 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 a- Express the function F in terms of S.O.P. b- Verify that F can be minimized to F = B'D' + BD, by using any method you want c- Realize the simplified function using NAND gates only. d-Realize the simplified function using NOR gates only. lo lo le

Simplify the following Boolean functions using four-variable maps. PLEASE EXPLAIN IN WRITING THE STEPS AND WHAT IS GOINING ON F(A, B, C, D) =∑ (4, 6, 7, 15) F(A, B, C,D) =∑ (0, 2, 4, 5, 6, 7, 8, 10, 13, 15)