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Please do parts a and b
A3. Let n be a positive integer and wadaa-cos (2π/n)+...
Ok = (6) Let n be a positive integer. For every integer k, define the 2...
Ok = (6) Let n be a positive integer. For every integer k, define the 2 x 2 matrix cos(27k/n) - sin(2nk/n) sin(2tk/n) cos(27 k/n) (a) Prove that go = I, that ok + oe for 0 < k < l< n - 1, and that Ok = Okun for all integers k. (b) Let o = 01. Prove that ok ok for all integers k. (c) Prove that {1,0,0%,...,ON-1} is a finite abelian group of order n.
8. Let w cos(2π/5) + isin(2π/5). Here we describe how to express w in terms of...
8. Let w cos(2π/5) + isin(2π/5). Here we describe how to express w in terms of square roots. (a) Show that w is a root of the polynomial 24+2+22+21. Hint: 25-1-(-(24+23+22+2+1) (b) Show that w + is a root of the polynomial u2 + u-1 (c) Show that Ve, where V5 means the positive square root of Hint: Figure out the sign of w by adding the polar forms of w and 1/w. (d) Put β--12vS So in part (c),...
5. Let Q-11,2,3,... be the countably infinite sample space whose elements (outcomes) are the positive integers....
5. Let Q-11,2,3,... be the countably infinite sample space whose elements (outcomes) are the positive integers. For each positive integer n, define the event An k k is a multiple of n ) (a) Find n and m such that An A3n A4 and Am A6n A9. (b) If P(tk))(3nd the probability of the event A3. k-1 Note: an exact answer is required here; if you write a program to obtain answers of the form 0.210526... you will receive no...
3 For each positive integer n, define E(n) 2+4++2n (a) Give a recursive definition for E(n)....
3 For each positive integer n, define E(n) 2+4++2n (a) Give a recursive definition for E(n). (b) Let P(n) be the statement E(n) nn1)." Complete the steps below to give a proof by induction that P(n) holds for every neZ+ i. Verify P(1) is true. (This is the base step.) ii. Let k be some positive integer. We assume P(k) is true. What exactly are we assuming is true? (This is the inductive hypothesis.) iii. What is the statement P(k...
Problem 4. Let w be a positive continuous function and let n be a nonnegative integer....
Problem 4. Let w be a positive continuous function and let n be a nonnegative integer. Equip P.(R) with the inner product (p,q) = $' p(x)q(x)"(x) dx. You do not need to check that this is an inner product. (a) Prove that P.(R) has an orthonormal basis po..., Pr such that deg pk = k for each k. (b) Show that (Pk, pk) = 0 for each k, where the polynomials pį are from the preceding part. Here pé denotes...
(a) Let A be a Hermitian matrix. DEFINE: A is positive definite. (b) Let A be an n × n Hermitian ...
(a) Let A be a Hermitian matrix. DEFINE: A is positive definite. (b) Let A be an n × n Hermitian matrix. PROVE: If A is positive definite the n every eigenvalue of A is positiv e. (c) Let Abe an n X n Hermitian matrix. PROVE: If every eigenvalue of A is positive. Then A is positive definite.
(a) Let A be a Hermitian matrix. DEFINE: A is positive definite. (b) Let A be an n × n Hermitian...
I got a C++ problem. Let n be a positive integer and let S(n) denote the...
I got a C++ problem.
Let n be a positive integer and let S(n) denote the number of divisors of n. For example, S(1)- 1, S(4)-3, S(6)-4 A positive integer p is called antiprime if S(n)くS(p) for all positive n 〈P. In other words, an antiprime is a number that has a larger number of divisors than any number smaller than itself. Given a positive integer b, your program should output the largest antiprime that is less than or equal...
* (9) Let n be a positive integer. Define : Z → Zn by (k) =...
* (9) Let n be a positive integer. Define : Z → Zn by (k) = [k]. (a) Show that is a homomorphism. (b) Find Ker(6) and Im(). yrcises (c) To what familiar group is the quotient group Z/nZ isomorphic? Explain.
QUESTION C. (a) Let k be a field and let n be a positive integer. Define what is meant by a monomial ideal in k[x,.....
QUESTION C.
(a) Let k be a field and let n be a positive integer. Define what is meant by a monomial ideal in k[x,...,zn]. 2. (b) State what it means for a ring R to be Noetherian. (c) State Hilbert's basis theorem. Give a proof of Hilbert's basis theorem using the fact if k is a field the polynomial ring kli,..., In] is Noetherian. 1S
(a) Let k be a field and let n be a positive integer. Define...
1. Let m be a nonnegative integer, and n a positive integer. Using the division algorithm...
1. Let m be a nonnegative integer, and n a positive integer. Using the division algorithm we can write m=qn+r, with 0 <r<n-1. As in class define (m,n) = {mc+ny: I,Y E Z} and S(..r) = {nu+ru: UV E Z}. Prove that (m,n) = S(n,r). (Remark: If we add to the definition of ged that gedan, 0) = god(0, n) = n, then this proves that ged(m, n) = ged(n,r). This result leads to a fast algorithm for computing ged(m,...
Ok = (6) Let n be a positive integer. For every integer k, define the 2 x 2 matrix cos(27k/n) - sin(2nk/n) sin(2tk/n) cos(27 k/n) (a) Prove that go = I, that ok + oe for 0 < k < l< n - 1, and that Ok = Okun for all integers k. (b) Let o = 01. Prove that ok ok for all integers k. (c) Prove that {1,0,0%,...,ON-1} is a finite abelian group of order n.
8. Let w cos(2π/5) + isin(2π/5). Here we describe how to express w in terms of square roots. (a) Show that w is a root of the polynomial 24+2+22+21. Hint: 25-1-(-(24+23+22+2+1) (b) Show that w + is a root of the polynomial u2 + u-1 (c) Show that Ve, where V5 means the positive square root of Hint: Figure out the sign of w by adding the polar forms of w and 1/w. (d) Put β--12vS So in part (c),...
5. Let Q-11,2,3,... be the countably infinite sample space whose elements (outcomes) are the positive integers. For each positive integer n, define the event An k k is a multiple of n ) (a) Find n and m such that An A3n A4 and Am A6n A9. (b) If P(tk))(3nd the probability of the event A3. k-1 Note: an exact answer is required here; if you write a program to obtain answers of the form 0.210526... you will receive no...
3 For each positive integer n, define E(n) 2+4++2n (a) Give a recursive definition for E(n). (b) Let P(n) be the statement E(n) nn1)." Complete the steps below to give a proof by induction that P(n) holds for every neZ+ i. Verify P(1) is true. (This is the base step.) ii. Let k be some positive integer. We assume P(k) is true. What exactly are we assuming is true? (This is the inductive hypothesis.) iii. What is the statement P(k...
Problem 4. Let w be a positive continuous function and let n be a nonnegative integer. Equip P.(R) with the inner product (p,q) = $' p(x)q(x)"(x) dx. You do not need to check that this is an inner product. (a) Prove that P.(R) has an orthonormal basis po..., Pr such that deg pk = k for each k. (b) Show that (Pk, pk) = 0 for each k, where the polynomials pį are from the preceding part. Here pé denotes...
(a) Let A be a Hermitian matrix. DEFINE: A is positive definite. (b) Let A be an n × n Hermitian matrix. PROVE: If A is positive definite the n every eigenvalue of A is positiv e. (c) Let Abe an n X n Hermitian matrix. PROVE: If every eigenvalue of A is positive. Then A is positive definite.
(a) Let A be a Hermitian matrix. DEFINE: A is positive definite. (b) Let A be an n × n Hermitian...
I got a C++ problem.
Let n be a positive integer and let S(n) denote the number of divisors of n. For example, S(1)- 1, S(4)-3, S(6)-4 A positive integer p is called antiprime if S(n)くS(p) for all positive n 〈P. In other words, an antiprime is a number that has a larger number of divisors than any number smaller than itself. Given a positive integer b, your program should output the largest antiprime that is less than or equal...
* (9) Let n be a positive integer. Define : Z → Zn by (k) = [k]. (a) Show that is a homomorphism. (b) Find Ker(6) and Im(). yrcises (c) To what familiar group is the quotient group Z/nZ isomorphic? Explain.
QUESTION C.
(a) Let k be a field and let n be a positive integer. Define what is meant by a monomial ideal in k[x,...,zn]. 2. (b) State what it means for a ring R to be Noetherian. (c) State Hilbert's basis theorem. Give a proof of Hilbert's basis theorem using the fact if k is a field the polynomial ring kli,..., In] is Noetherian. 1S
(a) Let k be a field and let n be a positive integer. Define...
1. Let m be a nonnegative integer, and n a positive integer. Using the division algorithm we can write m=qn+r, with 0 <r<n-1. As in class define (m,n) = {mc+ny: I,Y E Z} and S(..r) = {nu+ru: UV E Z}. Prove that (m,n) = S(n,r). (Remark: If we add to the definition of ged that gedan, 0) = god(0, n) = n, then this proves that ged(m, n) = ged(n,r). This result leads to a fast algorithm for computing ged(m,...
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