Suppose you are given an undirected, connected, weighted graph G. You want to find the heaviest weight set of edges that you can remove from G so that G is still connected. Propose an optimal greedy algorithm for this problem and prove that it is optimal. (Hint: use matroid theory!).

In question 1, do not use any properties of odd and even numbers outside of
their definitions. However, for questions 2 and 3, you can use the following
parity results without having to prove them each time.
For any integers a and b, the product a.b is odd if and only if a and
b are both odd.
For any integers, a and b, the sum a + b is even if and only if a and b
have the same parity.
For all questions, let x; y 2 Z.
1. Prove that if 11x - y is even, then x and y have the same parity.
2. Prove that 5xy + x + y is odd if and only if x or y is odd.
3. Prove that if 2|3x² - 1, then 4 |x² + 7.

A shipping company ShipCo supplies four destinations (D1,D2,D3,D4) from four sources (S1,S2,S3,S4). The shipping cost (in dollars) per shipment from each source to each destination is given below.

The four sources make 10, 20, 20, and 10 shipments per month, respectively. The four destinations need to receive 20, 10, 10, and 20 shipments per month, respectively. The manager of ShipCo now wants to determine the best plan for how many shipments to send from each source to each destination each month. The objective is to minimize the total shipping cost. Answer the following questions:
(a) Formulate this transportation problem as an LP. Clearly explain your variables and constraints.
(b) Use the Northwest corner method to find an initial BFS.
(c) Starting from the initial BFS in part (b), apply the transportation algorithm to solve this problem.

Let A be an n x n matrix satisfying A²=A (idempotent). Find all eigenvalues and eigenvectors of A.

I know that the eigenvalues are 0 and 1 -- I do not know how to find the eigenvectors.

a) Which variables are nonbasic, which ones are basic, and what are their respective values?

b) If I were minimizing the objective, which nonbasic variables are legitimate candidates to enter the basis? How about if I were maximizing?

c) Suppose I decided to enter x4 into the basis and increase its value by 100 units. Without doing any pivoting, can you say what the new objective value will be? Explain clearly.

d) Corresponding to this 100 unit increase in x4 what are the adjustments that need to be made to the values of the current basic variables in order to maintain feasibility? Explain clearly.

e) What is the maximum amount of increase possible in the value of x4? Explain clearly.

f) Again, without doing any pivoting, can you say what the value of the objective will be after the next iteration is completed?

Suppose $22,000 is invested at an annual rate of 5% for 20 years. Find the future value if interest is compounded as follows

a. Annually

b.Quarterly

c.Monthly

d.Daily (365 days)

e.Continuously

Congratulations, you just won the lottery! In one option presented to you, you will be paid one million dollars a year for the next 25 years. You can deposit this money in an account that will earn 5% each year.

a. Let M(t) be the amount of money in the account (measured in millions of dollars) at time t (measured in years). Set up a differential equation that describes the rate of change in the amount of money in the account. Two factors cause the amount to grow – first, you are depositing one million dollars per year and second, you are earning 5% interest.

b. The second option presented to you is to take a lump sum of 10 million dollars, which you will deposit into a similar account. Set up a new initial value problem (that is, differential equation with initial condition) to model this situation.

c. At what time does the amount of money in the account under the first option overtake the amount of money in the account under the second option?

a.Prove that {12a+ 4b | a, b ∈ Z}={4c |c ∈ Z}.

(b) Prove that{20a+ 16b|a, b ∈ Z}={28m+ 32n|m, n ∈ Z}.

(c) Leta, b ∈ Z−{0}. Prove that{x ∈ Z |ab divides x}⊆{x ∈ Z |a divides x}.

(d) Prove that{16n|n∈Z}⊆{2n|n ∈ Z}.

A particle of mass m is in a force field described by the equation,

                                 F(x) = bx

Here b is a positive constant. At t = 0, the particle is released from the origin (x₀ = 0) with an initial velocity V₀

a) Build a differential equation, and use a 'trial solution' to find the position of the particle as a function of time, x(t)

b) Use the 'separating variables and integration' method to find x(t)

schematize the following passage:

Ketogenic diet is well known for rapid weight loss. The diet protocol restricts carbohydrate consumption to 30 gm per day for an 80 kg adult. In order to protect the lean body mass, the protein intake should fall between 1.2-1.7 gm per kg BW per day, less than that may cause a loss of lean muscles. The protein intake should be moderate as an excessive amount of protein is metabolized to convert into amino acids, which are required for gluconeogenesis high-fat diet results in the shift of metabolism, depleted levels of glucose and enhanced oxidation of fatty acids that form ketone bodies to provide energy. The use of electrolytes such as sodium and potassium is highly recommended for maintaining nitrogen balance with the preservation of the fuctional tissue in the body.

Samreen Aziz, and Hina Rehman. “Mechanism and Benefits of Ketogenic Diet for Weight Loss and Health.” Rewal Medical Journal, vol. 44, no. 4, 2019, pp. 880–882., http://www.rmj.org.pk/fulltext/27-1545295565.pdf?1583710925.

1. For the initial-value problem, ?'(?) = ?? sin(5?) , ?(0) = 0.5,
2. Use the Euler Method with step size of 0.1 to calculate the approximate solution for the

differential equation for the initial condition y(0) = 0.5 and plot the result on the direction

field graph you produced in paragraph 1. Then do the same for the results of using the Euler Method with step sizes of 0.01 and then 0.005.

3. For the initial-value problem in paragraph 2 above, ?'(?) = ?? sin(5?) , ?(0) = 0.5,

find the approximate the value of y when x = 4 using the Euler Method for each of the three different step sizes you used in paragraph 2.

Type or write:
“For the Euler Method with a step size of 0.1, y(4) =”, followed by the answer accurate

to five decimal places.
“For the Euler Method with a step size of 0.01, y(4) =”, followed by the answer accurate

to five decimal places.
“For the Euler Method with a step size of 0.005, y(4) =”, followed by the answer accurate

to five decimal places.

4. Use the fourth-order Runge-Kutta Method with step sizes of 0.1, 0.01, and 0.005 to calculate approximate solutions for the same initial-value problem as paragraphs 2 and 3 and plot these approximate solutions on the direction field graph.
   • Attach a direction field graph showing the fourth-order Runge-Kutta Method results using a step size of 0.1.
   • Attach a second direction field graph showing the fourth-order Runge-Kutta Method results using a step size of 0.01.
   • Attach a third direction field graph showing the fourth-order Runge-Kutta Method results using a step size of 0.005.
   • Type or write a brief explanation of what happens with the plots as the step size is reduced.
5. For the initial-value problem in paragraphs 2 and 3 above, find the approximate the value of y when x = 4 using the fourth-order Runge-Kutta Method for each of the three different step sizes you used in paragraph 4.

Type or write:
“For the fourth-order Runge-Kutta Method with a step size of 0.1, y(4) =”, followed by

the answer accurate to five decimal places.
“For the fourth-order Runge-Kutta Method with a step size of 0.01, y(4) =”, followed by

the answer accurate to five decimal places.
“For the fourth-order Runge-Kutta Method with a step size of 0.005, y(4) =”, followed by

the answer accurate to five decimal places.

Prove that if p is a polynomial with degree n and k is a real number so that p(k) = 0,

then p(x)=(x-k)q(x) where q is a polynomial of degree n-1. Must use the fact that

a^n-b^n=(a-b)(a^(n-1) + a^(n-2)*b + ... + ab^(n-2) + b^(n-1)).

karla has opened an IRA in which she deposits $250 each month that earns 3.5% interest compounded monthly

how much money will she have in the account after 25 years?

how much total money will she put into the account?

how much total interest will she earn?

Melanie is the manager of the Clean Machine car wash and has gathered the following information. Customers arrive at a rate of eight per hour according to a Poisson distribution. The car washer can service an average of ten cars per hour with service times described by an ex- ponential distribution. Melanie is concerned about the number of customers waiting in line. She has asked you to calculate the following system characteristics:

(a) Average system utilization 8/10=0.8= 80%

(b) Average number of customers in the system 8/(10-8)=8/2= 4

(c) Average number of customers waiting in line 8^2/10(10-8)= 64/20= 3.2 customers

2. Melanie realizes that how long the customer must wait is also very important. She is also concerned about customers balking when the waiting line is too long. Using the arrival and service rates in Problem 1, she wants you to calculate the following system characteristics:

(a) The average time a customer spends in the system 1/(10-8)= ½ hours

(b) The average time a customer spends waiting in line0.8/(10-8)= 0.8/2 = 0.4 hours

(c) The probability of having more than three customers in the system 0.8^3 =0.512

(d) The probability of having more than four customers in the system 0.8^4 =0.4096

If Melanie adds an additional server at Clean Machine car wash, the service rate changes to an average of 16 cars per hour. The customer arrival rate is 10 cars per hour. Melanie has asked you to calculate the following system characteristics:
(a) Average system utilization
(b) Average number of customers in the system

(c) Average number of customers waiting in line

Melanie is curious to see the difference in waiting times for customers caused by the additional server added in Problem 3. Calculate the following system characteristics for her :

(a) The average time a customer spends in the system

(b) The average time a customer spends waiting in line

(c) The probability of having more than three customers in the system

(d) The probability of having more than four customers in the system

After Melanie added another car washer at Clean Machine (service rate is an average of 16 customers per hour), business improved. Melanie now estimates that the arrival rate is 12 customers per hour. Given this new information, she wants you to calculate the fol- lowing system characteristics:

(a) Average system utilization

(b) Average number of customers in the system

(c) Average number of customers waiting in line

As usual, Melanie then requested you to calculate sys- tem characteristics concerning customer time spent in the system.

(a) Calculate the average time a customer spends in the system.

(b) Calculate the average time a customer spends waiting in line.

(c) Calculate the probability of having more than four customers in the system.