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.

Prove that the union of a finite collection of compact subsets is compact

a) Solve IVP: y" + y' -2y = x + sin2x; y(0) = 1, y'(0) = 0

b) Solve using variation of parameters: y" -9y = x/e^3x

NUMBER THEORY QUESTION:

Find the partition of {1, 2, . . . , 16} determined by the dynamics of

(a) addition of 2, modulo 16.

(b) addition of 4, modulo 16

(c) multiplication by 2, modulo 17.

(d) multiplication by 4, modulo 17.

2. [6 marks] (Induction) Prove that 21 divides 4n+1 + 5 2n−1 whenever n is a positive integer. HINT: 25 ≡ 4(mod 21)

Takashi's family is a family of five of his parents, older brother, Takashi and younger brother. Currently, Takashi is half the age of his brother and his brother is half his age. The total age of the whole family is 117 years old. After three years, the total age of the parents will be twice the total age of the brothers, including Takashi.

At this time, how many years from now will it be that 1.5 times the age of my brother equals the sum of the age of Takashi and my brother? _____ years later

let takashi's old brother = k so takashi =k/2 and takashi younger brother= k/4 so (k+3) + (k+3/2)+ (k+3/4) + 2* (k+3) + (k+3/2)+ (k+3/4)= 117+3 is this right?

Question: Find √A for the following matrix A=[12 -3 8 ; 8 1 8 ; -3 3 1] then check if √A √A=A

Prove that none of the following is the order of a simple group: 28 , 12.

Thank you!

Please provide an example of a Monte Carlo simulation model (and explain).

Your simulation example should be able to:

1. Tackle a wide variety of problems using simulation
2. Understand the seven steps of conducting a simulation
3. Explain the advantages and disadvantages of simulation
4. Develop random number intervals and use them to generate outcomes
5. Understand alternative computer simulation packages available
6. Explain the different type of simulations
7. Explain the basic concept of simulation