Homework Answers
Please answer the parts 6 and 7. Thank you. 2. In this problem, we will prove...
Please answer the parts 6 and 7. Thank you.
2. In this problem, we will prove the following result: If G is a group of order 35, then G is isomorphic to Zg We will proceed by contradiction, so throughout the following questions, assume that G is a group of order 35 that is not cyclic. Most of these questions can he solved independently I. Show that every element of G except the identity has order 5 or 7. Let...
Let G be a group of order 6 and let X be the set (a, b,c) E G3: abc That is, X is the set of trip...
Let G be a group of order 6 and let X be the set (a, b,c) E G3: abc That is, X is the set of triples of elements of G with the product of its coordinates equals the identity element of G (a) How many elements does X have? Hint: Every triple (a, b, c) in X is completely determined by the choice of a and b. Because once you choose a and b then c must be (ab)-1...
3. Let E be the elliptic curve y2-x3+x 6 over ZI1 1) Find all points on E by calculating the quad...
3. Let E be the elliptic curve y2-x3+x 6 over ZI1 1) Find all points on E by calculating the quadratic residues like the one demonstrated in the lecture 2) What is the order of the group? [Do not forget the identity element 0] 3) Given a point P - (2, 7), what is 2P? [point doubling] 4) Given another point Q (3, 6), what is PQ? [point addition]
3. Let E be the elliptic curve y2-x3+x 6 over ZI1...
1 1 point Consider the following algorithm for factoring an integer N provided as input (in...
1 1 point Consider the following algorithm for factoring an integer N provided as input (in binary): For i = 2 to [VN.17 i divides N, then output (i, N/). Which of the following statements is true? This algorithm is correct, but it runs in exponential time. This algorithm is not correct, because it will sometimes fail to find a factorization of Neven if N is composite This algorithm runs in sub-linear time, and always factors N it Nis composite...
(6 points) Let X and Y be independent random variables with p.d.f.s fx(x) -{ { 1-22...
(6 points) Let X and Y be independent random variables with p.d.f.s fx(x) -{ { 1-22 0, for |2|<1, otherwise. fy(y) = for y>0, otherwise. 0, Let W = XY (a) (2 points) Find the p.d.f. of W, fw(w). (b) (2 points) Find the moment generating function of W2, Mw?(t) = E (e«w?). (c) (2 points) Find the conditional expectation of W given Y = y, E(W|Y = y).
3. Let G be a group containing 6 elements a, b, c, d, e, and f....
3. Let G be a group containing 6 elements a, b, c, d, e, and f. Under the group operation called the multiplication, we know that ad = c, bd = f, and f2 = bc = e. We showed you in class that the identity is e, hence the e-row and e-column were revealed. Using associativity, we also found cb, cf, af, and a2. Now try to imitate the idea and find five more entries. Justify your answer. Hint:...
Q9 6. Define Euclidean domain. 7. Let FCK be fields. Let a € K be a...
Q9
6. Define Euclidean domain. 7. Let FCK be fields. Let a € K be a root of an irreducible polynomial pa) EFE. Define the near 8. Let p() be an irreducible polynomial with coefficients in the field F. Describe how to construct a field K containing a root of p(x) and what that root is. 9. State the Fundamental Theorem of Algebra. 10. Let G be a group and HCG. State what is required in order that H be...
just answer e through h 8. (11 pts) Let (Xn) be a sequence in Rº such...
just answer e through h
8. (11 pts) Let (Xn) be a sequence in Rº such that VnEN, Xn+1 = A· Xn+ where A = (5/8 5/3) and Xo = (-1) (a) (1 pt) Find X1. (b) (2 pts) Find the corresponding equilibrium point. (c) (1 pt) Determine the two eigenvalues 11 and 12 of A. (d) (1 pt) For each eigenvalue, find an eigenvector. (e) (1 pt) Is the equilibrium point a sink? Justify. (f) (1 pt) Deduce the...
e. 18 Test Your Knowledge MULTIPLE CHOICE: Choose the one best answer. 1. Each element has...
e. 18 Test Your Knowledge MULTIPLE CHOICE: Choose the one best answer. 1. Each element has its own characteristic atom in which a. the atomic mass is constant. b. the atomic number is constant. c. the mass number is constant. d. Two of the above are correct. e. All of the above are correct. 2. Which of the following is not a trace element in the human body? a. iodine b. zinc c. iron d. calcium e. fluorine 3. A...
Please answer the parts 6 and 7. Thank you.
2. In this problem, we will prove the following result: If G is a group of order 35, then G is isomorphic to Zg We will proceed by contradiction, so throughout the following questions, assume that G is a group of order 35 that is not cyclic. Most of these questions can he solved independently I. Show that every element of G except the identity has order 5 or 7. Let...
Let G be a group of order 6 and let X be the set (a, b,c) E G3: abc That is, X is the set of triples of elements of G with the product of its coordinates equals the identity element of G (a) How many elements does X have? Hint: Every triple (a, b, c) in X is completely determined by the choice of a and b. Because once you choose a and b then c must be (ab)-1...
3. Let E be the elliptic curve y2-x3+x 6 over ZI1 1) Find all points on E by calculating the quadratic residues like the one demonstrated in the lecture 2) What is the order of the group? [Do not forget the identity element 0] 3) Given a point P - (2, 7), what is 2P? [point doubling] 4) Given another point Q (3, 6), what is PQ? [point addition]
3. Let E be the elliptic curve y2-x3+x 6 over ZI1...
1 1 point Consider the following algorithm for factoring an integer N provided as input (in binary): For i = 2 to [VN.17 i divides N, then output (i, N/). Which of the following statements is true? This algorithm is correct, but it runs in exponential time. This algorithm is not correct, because it will sometimes fail to find a factorization of Neven if N is composite This algorithm runs in sub-linear time, and always factors N it Nis composite...
(6 points) Let X and Y be independent random variables with p.d.f.s fx(x) -{ { 1-22 0, for |2|<1, otherwise. fy(y) = for y>0, otherwise. 0, Let W = XY (a) (2 points) Find the p.d.f. of W, fw(w). (b) (2 points) Find the moment generating function of W2, Mw?(t) = E (e«w?). (c) (2 points) Find the conditional expectation of W given Y = y, E(W|Y = y).
Q9
6. Define Euclidean domain. 7. Let FCK be fields. Let a € K be a root of an irreducible polynomial pa) EFE. Define the near 8. Let p() be an irreducible polynomial with coefficients in the field F. Describe how to construct a field K containing a root of p(x) and what that root is. 9. State the Fundamental Theorem of Algebra. 10. Let G be a group and HCG. State what is required in order that H be...
just answer e through h
8. (11 pts) Let (Xn) be a sequence in Rº such that VnEN, Xn+1 = A· Xn+ where A = (5/8 5/3) and Xo = (-1) (a) (1 pt) Find X1. (b) (2 pts) Find the corresponding equilibrium point. (c) (1 pt) Determine the two eigenvalues 11 and 12 of A. (d) (1 pt) For each eigenvalue, find an eigenvector. (e) (1 pt) Is the equilibrium point a sink? Justify. (f) (1 pt) Deduce the...
e. 18 Test Your Knowledge MULTIPLE CHOICE: Choose the one best answer. 1. Each element has its own characteristic atom in which a. the atomic mass is constant. b. the atomic number is constant. c. the mass number is constant. d. Two of the above are correct. e. All of the above are correct. 2. Which of the following is not a trace element in the human body? a. iodine b. zinc c. iron d. calcium e. fluorine 3. A...
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