#13. Design Turing machines to compute the following functions for x and y positive integers represented in unary: (Use Jflap to construct and test the TM) (b) f(x, y) = x + 2y.

The following equation can be used to compute values of y as a function of x: y=be¯ªsin(bx) (0.0012x¹ −0.15x³ +0.075x² +2.5x) where a and b are parameters. (a) Write Python code to compute a vector y for a = 2, b = 5, and x as a vector with values from 0 to π/2 in increments Ax= π/40. (b) Compute the vector z = y² with element-by-element-wise operations. (c) Combine x, y, and zinto a 3-column matrix W.(Hint use np. hstack function) (d) Generate a plot of y and z versus x. Include a title, axis labels, and a legend. For the y versus x curve, format a solid blue line. For the z versus x curve, choose a magenta dashed line. Include grid lines.

(1 point) Consider the points which satisfy the equation 3 y² = x² + ax + b mod p where a = 5, 61, and p = 7. Enter a comma separated list of points (x, y) consisting of all points in Z½-½ satisfying the equation. (Do not try to enter O, the point at infinity.) (0,1), (0.6),(1,0), (3,1), What is the cardinality of this elliptic curve group? 12

The following equation can be used to compute values of y as a function of x: y=be¯ªsin(bx) (0.0012x¹ −0.15x³ +0.075x² +2.5x) where a and b are parameters. (a) Write Python code to compute a vector y for a = 2, b = 5, and x as a vector with values from 0 to π/2 in increments Ax= π/40. (b) Compute the vector z = y² with element-by-element-wise operations. (c) Combine x, y, and zinto a 3-column matrix W.(Hint use np. hstack function) (d) Generate a plot of y and z versus x. Include a title, axis labels, and a legend. For the y versus x curve, format a solid blue line. For the z versus x curve, choose a magenta dashed line. Include grid lines.