Find all primes p for which −5 is a quadratic residue.

A=

Diagonalize the matrix above. That is, find matrix D and a nonsingular matrix P such that A = PDP^-1 . Use the representation to find the entries of Aⁿ as a function of n.

Let E/F be a field extension. Let a,b be elements elements of E and algebraic over F. Let m=[Q(a):Q] and n=[Q(b):Q]. Assume that gcd(m,n)=1. Determine the basis of Q(a,b) over Q.

2)

A university finished the season ranked sixth out of 119 teams in football and ninth out of 297 teams in baseball. Based on percentile rank, which team had the better ranking?

A)

the baseball team

Please explain

B)

the football team

Please give a response to below paragraph the one marked response not the question.

The Question:Discuss the properties of some of the commonly used functions on the set of Integers and on the set of Real Numbers. (Ex. Is the exponentiation any or none of Injective, Surjective, Bijective?, etc.)

Response: An exponential function such as exp(x) = e^x is an example of a commonly used injective function, which is a function that is one-to-one, meaning the elements of the domain are not mapped to the same codomain. The example, e^x is not surjective meaning, that the function does not have a right inverse (exp(x)^-1= e^x), in other words, every point in the codomain needs to be mapped to the domain to be considered a surjective function. At first, I was a little confused about what a bijection was as the definition sounds similar to an injective function, but if I understand it right, a bijection refers to a one-to-one correspondence. The difference being that a bijection is a function that is injective and surjective at the same time, and the exponential example (exp(x)=e^x) is not a bijective function since it is injective but not surjective.

Let R^x denote the group of nonzero real numbers under multiplication and let R⁺ denote the group of positive real numbers under multiplication. Let H be the subgroup {1, −1} of R^x. Prove that R^x ≈ R⁺ ⊕ H.

For each given integer n, determine if n is a sum of two squares. (You do not have to find the squares.)

(a) n = 19 (b) n = 45 (c) n = 99 (d) n = 999 (e) n = 1000

a. Show that for all values of ? there is an infinite sequence of positive eigenvalues of the problem

? ′′(?) + ??(?) = 0

? ?(0) + ? ′ (0) = 0, ?(1) = 0 (? = ?????)

b. Find eigenvalues of the problem if ? = 1.

The payoff matrix for a game is 4 −1 5 −6 2 1 1 −4 2 . (a) Find the expected payoff to the row player if the row player R uses the maximin pure strategy and the column C player uses the minimax pure strategy. (b) Find the expected payoff to the row player if R uses the maximin strategy 40% of the time and chooses each of the other two rows 30% of the time while C uses the minimax strategy 50% of the time and chooses each of the other two columns 25% of the time. (Round your answer to two decimal places.)

For each of the following degree sequences, determine if the exists a graph whose degree sequence is the one specified. In each case, either draw a graph or explain why no such graph exists.

a. (5,4,3,2,1)

b. (5,4,3,3,2,1)

c. (5,5,4,3,2,1)

Please show work - Discrete Mathematics - THANKS

Work out the distance of the point [1, 0, 0, 0] from the subspace W = ker T, where T : R 4 → R 2 is the linear map with matrix A = {(1,2,2,0), (2,1,3,1)}

Solve this differential equation using Matlab

yy' + xy² =x , with y(0)=5 for x=0 to 2.5 with a step size 0.25

(a) Analytical

(b) Euler

(c) Heun

d) 4th order R-K method

Display all results on the same graph

Prove that there are domains R containing a pair of elements havig no gcd.(Hint: let R be a subring of F[x], for a field F, where R consistes of all polynomials with no linear terms, then show that x⁵ and x⁶ have no gcd) .

7. Reduce the following expressions mod 10. No calculators! Efficiency matters. a)1212345670 ∙ 189000076548906789 ∙ 1234501 ∙ 12345678917 b) 123456789 + 987654321 + 22222222222222 + 8679

1. Write down the addition and multiplication table for Z/5Z. All classes should be written in terms of their canonical representative (unique representative between 0 and 4).

2. Suppose a ≡ a' mod n and b ≡ b' mod n.

(a) Show that a + b ≡ a' + b' mod n.

(b) Show that a · b ≡ a' · b' mod n. (An important consequence of this exercise is that addition and multiplication define maps Z/nZ × Z/nZ → Z/nZ. This is not obvious from the definition of addition and multiplication, because the definitions require you to choose representatives for equivalence classes. Different choices of representatives could conceivably result in different outcomes for the sum or product of the same pair of elements. A map, however, is not allowed to send the same pair of elements to multiple distinct elements. For example, in Z/123Z, we have [10] + [5] = [10 + 5] = [15] by definition. However, [10] = [625] and [5] = [866]. Applying the definition of sum again yields [625] + [866] = [1491]. If addition does define a map, then it should be the case that [15]=[1491], otherwise the pair ([10],[5]) would be mapped to two distinct values. A simple check shows that this is indeed the case here, and this exercise shows that this will be true in general.)

3. We extend the notion of divisibility to Z/nZ in the obvious way: [a]|[b] if [b] = [k] · [a] for some k ∈ Z.

(a) Prove that [a]|[b] if and only if gcd(a, n)|b. (Hint: use problem 1: Let a, b, c ∈ Z. Prove that c = ma + nb for some m, n ∈ Z if and only if gcd(a, b)|c.)

(b) Conclude from part (a) that [a] has a multiplicative inverse if and only if gcd(a, n) = 1.