Prove that the number of elements of order n in Zn is exactly ?(n), the Euler...
Prove that the number of elements of order n in Zn is exactly ?(n), the Euler phi function of n. Hint: You need to decide which [a] ? Zn generate Zn.
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Prove that the number of elements of order n in Zn is exactly
?(n), the Euler...
Use the Euler-Totient Function formula to find all values n, such that phi(n)<=5. Prove that t...
Use the Euler-Totient Function formula to find all values n, such that phi(n)<=5. Prove that the list is indeed correct.
Part 15A and 15B (15) Let n E Z+,and let d be a positive divisor of...
Part 15A and 15B
(15) Let n E Z+,and let d be a positive divisor of n. Theorem 23.7 tells us that Zn contains exactly one subgroup of order d, but not how many elements Z has of order d. We will determine that number in this exercise. (a) Determine the number of elements in Z12 of each order d. Fill in the table below to compare your answers to the number of integers between 1 and d that are...
(3.5) Summing the Euler S-function (n): The Euler 6-function 6(n) counts the number of positive integers...
(3.5) Summing the Euler S-function (n): The Euler 6-function 6(n) counts the number of positive integers less than or equal to n, which are relatively prime with n. Evaluate 4(d), and prove that your answer is correct. (3.4) Relatively Prime Numbers and the Chinese Re- mainder Theorem: Give an example of three positive integers m, n, and r, and three integers a, b, and c such that the GCD of m, n, and r is 1, but there is no...
Prove that Z/ ≡3 has exactly three elements using the given hint! Definition: Let R be...
Prove that Z/ ≡3 has exactly three elements using the
given hint!
Definition: Let R be an equivalence relation on the set A. The set of all equivalence classes is denoted by A/R (g) Prove that Z/ has exactly three elements. Hint: First, verify that [5]3, [7]3, and [013 are three different elements of Z/-3-Then, verify that every m E Z is in one of these sets. Then explain why those two facts imply that [5]3, [7 3, and [013...
Prove that the number of unordered sequences of length k with elements from a set X...
Prove that the number of unordered sequences of length k with elements from a set X of size n is n+k−1 k . Hint: For illustration, first consider the example n = 4, k = 6. Let the 4 elements of the set X be denoted a, b, c, d. Argue that any unordered sequence of size 6 consisting of elements a, b, c, d can be represented uniquely by a symbol similar to “··|·|··|·”, corresponding to the sequence aabccd....
Suppose that (Wn: n 2 0) is an autoregressive sequence of order 1, so that for n 2 0, where the Z...
Suppose that (Wn: n 2 0) is an autoregressive sequence of order 1, so that for n 2 0, where the Z's are i.i.d. and independent of Wo. a) Express Wn as a function of Wo, Z1, , Zn Suppose that |ρ| 〈 1 and var(WD+var(ZI) 〈 OO b) Compute cov(Wm, Wn) for m, n 2 0 c) Prove that there exists a deterministic constant a for which as n -oo and compute a. (Hint: Compute var(Wn)) Suppose, in addition,...
Use MATLAB to prove the central limit theorem. To achieve this, you will need to generate N rando...
Use MATLAB to prove the central limit theorem. To achieve this, you will need to generate N random variables (I.I.D. with the distribution of your choice) and show that the distribution of the sum approaches a Guassian distribution. Plot the distribution and matlab code. Hint: you may find the hist) function helpful.
a) Prove algebraically that(m+n | p+n)≥(m | p) for all m, p, n ∈ N and...
a) Prove algebraically that(m+n | p+n)≥(m | p) for all m, p, n ∈
N and such that m≥p.
b) Prove the above inequality by providing a combinatorial
proof. Hint: this can be done by creating a story to count the RHS
exactly (and explain why that count is correct), and then providing
justification as to why the LHS counts a larger number of
options.
a) Prove algebraically that p for all m, p, n EN, and such that m...
2. Count the number N(k) of all finite automata with exactly k states. Prove that in...
2. Count the number N(k) of all finite automata with exactly k states. Prove that in the definition of the regularity we can't restrict the num- ber or states to be less than some fixed integer
(4) Let p Z be a prime. Prove that zli/(p+1) has exactly ] p2 +1 elements. Use that 5+5i (2+i)(3+...
(4) Let p Z be a prime. Prove that zli/(p+1) has exactly ] p2 +1 elements. Use that 5+5i (2+i)(3+i) to determine how many elements Zu/5+5i) has. (5) Let m,n be integers with m|n. Prove that the surjective ring homomor- phism Z/n -> Z/m induces a group homomorphism on units, and that this group homomorphism is also surjective.
(4) Let p Z be a prime. Prove that zli/(p+1) has exactly ] p2 +1 elements. Use that 5+5i (2+i)(3+i) to determine...
Part 15A and 15B
(15) Let n E Z+,and let d be a positive divisor of n. Theorem 23.7 tells us that Zn contains exactly one subgroup of order d, but not how many elements Z has of order d. We will determine that number in this exercise. (a) Determine the number of elements in Z12 of each order d. Fill in the table below to compare your answers to the number of integers between 1 and d that are...
(3.5) Summing the Euler S-function (n): The Euler 6-function 6(n) counts the number of positive integers less than or equal to n, which are relatively prime with n. Evaluate 4(d), and prove that your answer is correct. (3.4) Relatively Prime Numbers and the Chinese Re- mainder Theorem: Give an example of three positive integers m, n, and r, and three integers a, b, and c such that the GCD of m, n, and r is 1, but there is no...
Prove that Z/ ≡3 has exactly three elements using the
given hint!
Definition: Let R be an equivalence relation on the set A. The set of all equivalence classes is denoted by A/R (g) Prove that Z/ has exactly three elements. Hint: First, verify that [5]3, [7]3, and [013 are three different elements of Z/-3-Then, verify that every m E Z is in one of these sets. Then explain why those two facts imply that [5]3, [7 3, and [013...
Suppose that (Wn: n 2 0) is an autoregressive sequence of order 1, so that for n 2 0, where the Z's are i.i.d. and independent of Wo. a) Express Wn as a function of Wo, Z1, , Zn Suppose that |ρ| 〈 1 and var(WD+var(ZI) 〈 OO b) Compute cov(Wm, Wn) for m, n 2 0 c) Prove that there exists a deterministic constant a for which as n -oo and compute a. (Hint: Compute var(Wn)) Suppose, in addition,...
a) Prove algebraically that(m+n | p+n)≥(m | p) for all m, p, n ∈
N and such that m≥p.
b) Prove the above inequality by providing a combinatorial
proof. Hint: this can be done by creating a story to count the RHS
exactly (and explain why that count is correct), and then providing
justification as to why the LHS counts a larger number of
options.
a) Prove algebraically that p for all m, p, n EN, and such that m...
2. Count the number N(k) of all finite automata with exactly k states. Prove that in the definition of the regularity we can't restrict the num- ber or states to be less than some fixed integer
(4) Let p Z be a prime. Prove that zli/(p+1) has exactly ] p2 +1 elements. Use that 5+5i (2+i)(3+i) to determine how many elements Zu/5+5i) has. (5) Let m,n be integers with m|n. Prove that the surjective ring homomor- phism Z/n -> Z/m induces a group homomorphism on units, and that this group homomorphism is also surjective.
(4) Let p Z be a prime. Prove that zli/(p+1) has exactly ] p2 +1 elements. Use that 5+5i (2+i)(3+i) to determine...
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