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

?′ (?) = −100?(?), ?(0) = y₀

1) Find the condition for the step size dt such that the explicit Euler scheme converges to the exact solution to the given differential equation.

2) Find the condition for the step size dt such that the implicit Euler scheme converges to the exact solution to the given differential equation.

Verify Stokes theorem for F =(y^2 + x^2 - x^2)i + (z^2 + x^2 - y^2)j + (x^2 + y^2 - z^2)k over the portion of the surface x^2 + y^2 -2ax + az = 0

Solve the linear second-order ODE for each case of b. Find constants using the given initial conditions.

y(0)=1, y'(0)=0

y''+by'+16y=0

b=0

b=2

b=8

b=10

say b represents damping constant. What is the effect of damping on the motion of a mass?

1.Describe all of the homomorphisms from Z20 to Z40.

2.Describe all of the homomorphisms from Z to Z12.

A poker company assembles three different poker sets. Each Royal Flush poker set contains 1000 poker​ chips, 4 decks of​ cards, 10 ​dice, and 2 dealer buttons. Each Deluxe Diamond poker set contains 600 poker​ chips, 2 decks of​ cards, 5 ​dice, and one dealer button. The Full House poker set contains 300 poker​ chips, 2 decks of​ cards, 5 ​dice, and one dealer button. The company has 2 comma 900,000 poker​ chips, 10,000 decks of​ cards, 25,000 ​dice, and 6000 dealer buttons in stock. They earn a profit of ​$38 for each Royal Flush poker​ set, ​$22 for each Deluxe Diamond poker​ set, and ​$12 for each Full House poker set. Complete parts​ (a) and​ (b) below.

​(a) How many of each type of poker set should they assemble to maximize​ profit? What is the maximum​ profit?

Begin by finding the objective function. Let x 1x1 be the number of Royal Flush poker​ sets, let x 2x2 be the number of Deluxe Diamond poker​ sets, and let x 3x3 be the number of Full House poker sets. What is the objective​ function?

Prove that the product of a finite number of compact spaces is compact.