1. Let Xi, i = 1, 2,. ..., n + 1 different nodes and let yį € R, i = 1,2,. interpolating polynomial is written in Newton's form as: Pn(x) = = a₁ + a₂(x-x₁) + az(x − x₁)(x - x₂)+ ·+an+1(x-x₁)... (x − xn+1), where the coefficients a¿, i = 1, ..., n + 1 can be computed using the following algorithm: Algorithm 1 Newton's polynomial aiyi, i = 1,2,...,n+1 for k= 2: n + 1 do for i=1: k- 1 do ak = = (ak — α₂)/(xk — Xi) end for end for If the coefficients ai, i = 1,..., n+1 are known, then the value of the interpolating polynomial at the point z can be computed using Horner's formula: Algorithm 2 Horner's formula S = an+1 for in-1:1 do s = a₁ + (2x₁) s end for Pn (2) = ,n+ 1. The = S Remark: It is noted that in the loop conditions i= a b c of the previous pseudo-codes a is the starting value, b is the step and c is the last value. (a) Write PYTHON 's functions coefs and evalp implementing the previously described algorithms for the coefficients of the interpolating polynomial and it's evaluation at values z using Horner's formula. Write a function newtinterp with input arguments (x, y, z), where (xi, Yi), i = 1,..., n+1, the interpolation data points and z = [21, ..., Zm] the vector containing the m points on which we want to evaluate the interpolating polynomial. This will compute the coefficients of the interpolating polynomial a; using your function coefs and will return the values uį = Pn (zi), i=1,2,...,m using your function evalp. (b) Consider the function f(x) = sinx, x = [0, 2π], and a uniform partition of [0, 2] with 6 points xi, i = 1,...,6. Using your function newtinterp compute the interpolating polynomial p5, that interpolates function ƒ at the nodes ï¿, at 101 equi-distributed points Zį € [0, 2π].

Let E : y2 = x3 + 3x + 4 be an elliptic curve over F37. (a) Find all the elements of the elliptice curve group.(b) Find the order of the group.(c) Find a primitive element of this group and call it G.(d) Compute [30]G = G ⊕ G ⊕ · · · ⊕ G (addition of 30 many G’s)   kindly help with full explanation. Thank you!

For all positive numbers aaand bbwith a>ba>b, ln⁡(a−b)=ln⁡(a)/ln⁡(b)ln(a−b)=ln(a)/ln(b) True or false

The question describes a function S(k) which is defined as the sum of the positive divisors of a positive integer k, minus k itself. The function S(1) is defined as 1, and for any positive integer k greater than 1, S(k) is calculated as S(k) = σ(k) - k, where σ(k) is the sum of all positive divisors of k. Some examples of S(k) are given: S(1) = 1 S(2) = 1 S(3) = 1 S(4) = 3 S(5) = 1 S(6) = 6 S(7) = 1 S(8) = 7 S(9) = 4 The question then introduces a recursive sequence a_n with the following rules: a_1 = 12 For n ≥ 2, a_n = S(a_(n-1)) Part (a) of the question asks to calculate the values of a_2, a_3, a_4, a_5, a_6, a_7, and a_8 for the sequence. Part (b) modifies the sequence to start with a_1 = k, where k is any positive integer, and the same recursion formula applies: for n ≥ 2, a_n = S(a_(n-1)). The question notes that for many choices of k, the sequence a_n will eventually reach and remain at 1, but this is not always the case. It asks to find, with an explanation, two specific…