![If Xo = 3 and Xn = (5xn-1 + 7) mod 200 find x], ..., X 10.](http://img.homeworklib.com/questions/206e0640-9a73-11eb-9be5-d914c08874d6.png?x-oss-process=image/resize,w_560)
Homework Answers
9. Use the construction in the proof of the Chinese remainder theorem to find a solution to the system of congruences X 1 mod 2 x 2 mod 3 x 3 mod 5 x 4 mod 11 10. Use Fermats little theorem to fin...
9. Use the construction in the proof of the Chinese remainder theorem to find a solution to the system of congruences X 1 mod 2 x 2 mod 3 x 3 mod 5 x 4 mod 11 10. Use Fermats little theorem to find 712 mod 13 11. What sequence of pseudorandom numbers is generated using the linear congruential generator Xn+1 (4xn + 1) mod 7 with seed xo 3?
9. Use the construction in the proof of the Chinese...
11. What sequence of pseudorandom numbers is generated using the linear congruential generator xn +1 (4xn + 1) mod 7 with seed Xo-37 12. Encrypt the message STOP POLLUTION by translating the letters...
11. What sequence of pseudorandom numbers is generated using the linear congruential generator xn +1 (4xn + 1) mod 7 with seed Xo-37 12. Encrypt the message STOP POLLUTION by translating the letters into numbers, applying the encryption function/ P)-(p + 4) mod 26, and then translating the numbers back into letters. 13. Decrypt this message encrypted using the shift cipher f (p) (p+ 10) mod 26 CEBBOXNOBXYG 14. Let P() be the statement that 12 +22 ++n2 -n-)(en+2) for...
[ARCHIMEDES] Suppose that Xo = 2/3, yo = 3, xn = 2xn-1 Yn-1 xn-1 + Yn-1...
[ARCHIMEDES] Suppose that Xo = 2/3, yo = 3, xn = 2xn-1 Yn-1 xn-1 + Yn-1 and Yn = 1xn Yn-1 for ne N. a) Prove that xnx and Yn 1 y, as n = , for some x, y E R. b) Prove that x = y and 3.14155 < x < 3.14161. (The actual value of x is a.)
2. Let Xn,n0,1,2,... denote a biased random walk given by Xo 0 and Xn+1 Xn + YTHI, where (X } are...
2. Let Xn,n0,1,2,... denote a biased random walk given by Xo 0 and Xn+1 Xn + YTHI, where (X } are 1.1.d. random variables with N(-1,1) distribution. Show that Mn X22n Xn (n -1) is a martingale.
2. Let Xn,n0,1,2,... denote a biased random walk given by Xo 0 and Xn+1 Xn + YTHI, where (X } are 1.1.d. random variables with N(-1,1) distribution. Show that Mn X22n Xn (n -1) is a martingale.
Xo Xo Problem 1. Show that the recursively-defined sequence x*i-x, - gives the sequence of x-valu...
Xo Xo Problem 1. Show that the recursively-defined sequence x*i-x, - gives the sequence of x-values described in this procedure, as follows: (a) Write the linear approximation 1 (x) to the curve at the point (Xn,f(xn). (b) Find where this linear approximation passes through the x-axis by solving L(x)0 for x. xn + 1-1,-I n). is the recursion formula for Newton's Method. :
Xo Xo Problem 1. Show that the recursively-defined sequence x*i-x, - gives the sequence of x-values described...
(a) Solve the simultaneous congruences p = 1 (mod x – 3), p = 7 (mod...
(a) Solve the simultaneous congruences p = 1 (mod x – 3), p = 7 (mod x – 5). (b) Find the total number of monic irreducible polynomials of degree 5 in Fr[c]. (c) Find a primitive root modulo 52020. (Make sure to justify your answer.) (d) Determine the total number of primitive roots modulo 52020.
a) b) Consider the linear congruential generator Xi41 = (5X; 1)mod(8). Using Xo 0, calculate the...
a)
b)
Consider the linear congruential generator Xi41 = (5X; 1)mod(8). Using Xo 0, calculate the 99th pseudo-random number Ug9 16807 X-1 mod(231 - 1). Using Consider our "desert island" PRN generator, X; Xo 12345678, calculate X99.
Consider the linear congruential generator Xi41 = (5X; 1)mod(8). Using Xo 0, calculate the 99th pseudo-random number Ug9
16807 X-1 mod(231 - 1). Using Consider our "desert island" PRN generator, X; Xo 12345678, calculate X99.
Find the smallest positive solution and the general solution to the system x ≡ 1 (mod...
Find the smallest positive solution and the general
solution to the system x ≡ 1 (mod 3), x ≡ 2 (mod 5) and x ≡ 3 (mod
7).
Exercise 2 (5 points Find the smallest positive solution and the general solution to the system ΧΞ2 (mod 5) and r Ξ 3 (mod 7). 1 (mod 3),
This is for Stochastic Processes Let Xo, Xi,... be a Markov chain whose state space is Z (the integers). Recall the Markov property: P(X, _ in l Xo-to, X1-21, , Xn l-an l)-P(Xn-in l x, i-İn 1), Vn, V...
This is for Stochastic Processes
Let Xo, Xi,... be a Markov chain whose state space is Z (the integers). Recall the Markov property: P(X, _ in l Xo-to, X1-21, , Xn l-an l)-P(Xn-in l x, i-İn 1), Vn, Vil. Does the following always hold: (lProve if "yes", provide a counterexample if "no")
Let Xo, Xi,... be a Markov chain whose state space is Z (the integers). Recall the Markov property: P(X, _ in l Xo-to, X1-21, , Xn l-an l)-P(Xn-in...
1. Prove for any Xo E R that the iteration In+1 = g(xn) converges to a...
1. Prove for any Xo E R that the iteration In+1 = g(xn) converges to a unique fix point a where g(x) = cos X. Find the value a to at least 14 decimal places.
9. Use the construction in the proof of the Chinese remainder theorem to find a solution to the system of congruences X 1 mod 2 x 2 mod 3 x 3 mod 5 x 4 mod 11 10. Use Fermats little theorem to find 712 mod 13 11. What sequence of pseudorandom numbers is generated using the linear congruential generator Xn+1 (4xn + 1) mod 7 with seed xo 3?
9. Use the construction in the proof of the Chinese...
11. What sequence of pseudorandom numbers is generated using the linear congruential generator xn +1 (4xn + 1) mod 7 with seed Xo-37 12. Encrypt the message STOP POLLUTION by translating the letters into numbers, applying the encryption function/ P)-(p + 4) mod 26, and then translating the numbers back into letters. 13. Decrypt this message encrypted using the shift cipher f (p) (p+ 10) mod 26 CEBBOXNOBXYG 14. Let P() be the statement that 12 +22 ++n2 -n-)(en+2) for...
[ARCHIMEDES] Suppose that Xo = 2/3, yo = 3, xn = 2xn-1 Yn-1 xn-1 + Yn-1 and Yn = 1xn Yn-1 for ne N. a) Prove that xnx and Yn 1 y, as n = , for some x, y E R. b) Prove that x = y and 3.14155 < x < 3.14161. (The actual value of x is a.)
2. Let Xn,n0,1,2,... denote a biased random walk given by Xo 0 and Xn+1 Xn + YTHI, where (X } are 1.1.d. random variables with N(-1,1) distribution. Show that Mn X22n Xn (n -1) is a martingale.
2. Let Xn,n0,1,2,... denote a biased random walk given by Xo 0 and Xn+1 Xn + YTHI, where (X } are 1.1.d. random variables with N(-1,1) distribution. Show that Mn X22n Xn (n -1) is a martingale.
Xo Xo Problem 1. Show that the recursively-defined sequence x*i-x, - gives the sequence of x-values described in this procedure, as follows: (a) Write the linear approximation 1 (x) to the curve at the point (Xn,f(xn). (b) Find where this linear approximation passes through the x-axis by solving L(x)0 for x. xn + 1-1,-I n). is the recursion formula for Newton's Method. :
Xo Xo Problem 1. Show that the recursively-defined sequence x*i-x, - gives the sequence of x-values described...
(a) Solve the simultaneous congruences p = 1 (mod x – 3), p = 7 (mod x – 5). (b) Find the total number of monic irreducible polynomials of degree 5 in Fr[c]. (c) Find a primitive root modulo 52020. (Make sure to justify your answer.) (d) Determine the total number of primitive roots modulo 52020.
a)
b)
Consider the linear congruential generator Xi41 = (5X; 1)mod(8). Using Xo 0, calculate the 99th pseudo-random number Ug9 16807 X-1 mod(231 - 1). Using Consider our "desert island" PRN generator, X; Xo 12345678, calculate X99.
Consider the linear congruential generator Xi41 = (5X; 1)mod(8). Using Xo 0, calculate the 99th pseudo-random number Ug9
16807 X-1 mod(231 - 1). Using Consider our "desert island" PRN generator, X; Xo 12345678, calculate X99.
Find the smallest positive solution and the general
solution to the system x ≡ 1 (mod 3), x ≡ 2 (mod 5) and x ≡ 3 (mod
7).
Exercise 2 (5 points Find the smallest positive solution and the general solution to the system ΧΞ2 (mod 5) and r Ξ 3 (mod 7). 1 (mod 3),
This is for Stochastic Processes
Let Xo, Xi,... be a Markov chain whose state space is Z (the integers). Recall the Markov property: P(X, _ in l Xo-to, X1-21, , Xn l-an l)-P(Xn-in l x, i-İn 1), Vn, Vil. Does the following always hold: (lProve if "yes", provide a counterexample if "no")
Let Xo, Xi,... be a Markov chain whose state space is Z (the integers). Recall the Markov property: P(X, _ in l Xo-to, X1-21, , Xn l-an l)-P(Xn-in...
1. Prove for any Xo E R that the iteration In+1 = g(xn) converges to a unique fix point a where g(x) = cos X. Find the value a to at least 14 decimal places.
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