Homework
Math 356
Spring 2007

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Course Schedule and a list of topics covered

Homework 15
due Friday, May 11, 2007
  • Read Sections 7.4 and 7.5.
    • Question 1: Use the simplex method to find the optimal solution and optimal value to 7.2.1.
    • Question 2: 7.5.2
    • Question 3: Explain how applying the Borda Count method with a choice vector with only zeroes and ones is different from approval voting, where each voter votes either yes or no for each candidate.
    • Question 4: The voters are using Borda Count to choose between four candidates, A, B, C, and D. Create a set of voters' preferences where three different candidates are elected when the choice vector [1,x,y,0] is varied. Draw the regions in the x-y plane where each candidate is elected.
    • Question 5: Just after the 2006 Canadian elections, the number of seats in the parliament were held by the following constituancies
      Conservative Liberal Bloc Qubcois NDP Independent
      124 102 51 29 1
      Soon thereafter, one elected official defected from the Liberal party to the Conservative Party, and the balance of power became
      Conservative Liberal Bloc Qubcois NDP Independent
      125 101 51 29 1
      Determine the Banzhaf power index for each party in both cases. Remember to show your work! Give a conceptual explanation for the change in party power between the two distributions. (Assume simple majority rule)   (Inspiration from "Power in Voting Games and Canadian Politics" by Chris Nicola.)
  • Bonus point possibility related to Question 4 (4 pts possible): Find the smallest number of voters necessary to elect each of the four candidates when the choice vector is varied. Prove that your number is optimal.
Homework 12
due Monday, April 16, 2007
  • Read Sections 5.5, 7.1, 7.2, and 7.3.
    • Question 1: Use a random walk simulation in Maple to determine the number of games (on average) it would take you to lose $20 in single-zero roulette betting red each game at $1 per game. You win $1 if you land on red (18 of 37 spaces) and lose $1 if you land on black or green (18+1=19 of 37 spaces). How much faster do you become bankrupt (on average) if you play double-zero roulette, where there are two green spaces?
    • Question 2: In the Maple lab, you modified the given boat queuing simulation to keep track of lost business if the length of the maximum queue is two. We'll call that harbor "Harbor A". Imagine that another harbor identical in every other respect has a maximum queue length of one. We'll call it "Harbor B". Using your simulation as a guide, would it make sense for Harbor B's owners to undertake an expansion project to change the maximum queue to two, just as with Harbor A? Address multiple concerns the owners might take into account.
    • ***If you are able to work out the modified simulation on your own, you can earn a bonus point. But the goal of this homework question is to use the simulation to do the analysis. To this end, I have modified the Maple worksheet taking the queue into account. You can find it on the Maple page or via this link directly.
    • Question 3: 7.1.5 (Answer the three questions at the bottom of page 244, and then write out the linear program mathematically.)
    • Question 4: 7.2.3 (Do it geometrically, by hand.)
    • Question 5: Write out the linear program that you need to solve in order to fit the following data to the curve y=cx2, and then use Maple to solve your linear program.
       x 7192530
       y 15120195305
  • An interesting extension to Question 1 (not part of this homework assignment) is to determine the average maximum amount of money you'll have during the game --- if you ever reach this value, you would probably decide to stop playing!
Homework 10
due Monday, March 26, 2007
  • Read Sections 4.3, 4.4, 5.0, 5.1, 5.2, and 5.3.
  • Make sure you check out the nifty flowchart on page 170
    • Question 1: (a) 4.3.3 (b) 4.3.4
    • Question 2: (a) Write out the equations to solve the cubic spline in 4.4.1a. (b) Use Maple to solve the system of equations, and compare your answer to Maple's spline command.
    • Question 3: 5.2.2abcd
    • Question 4: 5.1.4 (Then use Maple to calculate the volume through a Monte Carlo simulation.)
    • Question 5: 5.1.1 (And use Maple to simulate how many tries it will take you.)
Homework 8
due Monday, March 19, 2007
  • Read Sections 3.3, 3.4, 6.3, 4.1, and 4.2
  • The linear regression maple worksheet may be helpful on problems 1 and 4.
    • Question 1: 6.3.2
    • Question 2: (worth 8 points) the four questions from 4.1.1--4.1.4
    • Question 3: 4.2.3 or 4.2.4, your choice.
    • Question 4: Parenting as a science experiment: using the method of linear regression, determine if there is a correlation between baby milk consumption and length of sleep at night. Attached is a spreadsheet of our baby Chloe's milk consumption and feeding times since December. Do a linear regression between two variables, the first of which may be "the total amount of milk consumed in a day", "the amount of milk consumed in the last feeding", or "the percentage of total milk consumed in the last feeding", and the second of which may be "time that Chloe sleeps during the night" or "time that Chloe wakes up in the morning". You may assume that Chloe falls asleep 20 minutes after the last feeding and wakes up at the time of the first feeding. The milk quantity is in milliliters. You will need to be careful when you calculate times since sleeping from 9:30pm to 7:20am is not 12+7.20-9.30 hrs. That is, you should probably convert the times into minutes.
You might want to check out the exams page.
Homework 6 (Updated 24 Feb 5pm)
due Monday, February 26, 2007
  • Skim Sections 8.0, 8.1, and 8.2, and read Sections 3.0, 3.1, and 3.2.
  • Turn in well-written solutions to the following problems.
    • Question 1: (a) 8.1.2 (b) 8.1.4
    • Question 2: 8.2.11
    • Question 3: 3.1.1 (Also discuss Figure 3.3 as you did Figure 3.2.)
    • Question 4: 3.1.7
    • Question 5: 3.2.3 (You'll need to use Maple to solve for c1, c2, and c3; we'll do something similar in the lab on Thursday.) [Do this problem using least squares, not Chebyshev, as stated]
Term-Long Report Project Statement due Friday, February 23, 2007
Homework 5
due Monday, February 19, 2007
  • Read Sections 2.2, 2.3, and 2.4.
  • Turn in well-written solutions to the following problems.
    • Question 1: 2.2.6
    • Question 2: Here is some data that may support a proportionality argument.
       y 0.50.751.01.52.02.53.03.54.0
       z 0.090.270.622.155.010.317.627.240.9
      Use Maple to plot y versus z, z2, z3, and ez. For which, if any, of these relationships would a proportionality argument seem reasonable? You should print and discuss each graph.
    • Question 3: Problem 2.3.2
    • Question 4: Problem 2.3.4
    • Question 5: Project 2.3.4 [Important: this is NOT Problem 2.3.4. You just did it!]
Homework 4
due Monday, February 12, 2007
  • Read Sections 6.1, 2.0, and 2.1.
  • Turn in well-written solutions to the following problems.
    • Question 1: (Markov Chains) (a) 6.1.1 (b) 6.1.2
    • Question 2: (Linear Algebra) Consider the following system of difference equations.
      an+1 = -0.5an+2bn
      bn+1 = -an+2.5bn
      Determine the vector form of this set of difference equations. Find the eigenvalues λ1 and λ2 (those are lambdas) for the matrix A and their corresponding eigenvectors x1 and x2. (Choose each to be integral so that your calculations are nicer.) For each of the eigenvectors xi, determine the numerical solution to the system of difference equations with initial condition [a0,b0]T=xi. What is the numerical solution to the system if the initial condition is y=c1x1+c2x2 for constants c1 and c2? What (if any) are the equilibrium vectors in this system?
    • Question 3: Any one of Problems 2.1.1 through 2.1.8.
    • Question 4: Project 2.1.2 on p64 [Important: this is NOT Problem 2.1.2]
    • Question 5: (Maple) There is a link on the Maple page to Question 5
Homework 3 (Final Version, Revised 31 Jan Feb 1)
due Monday, February 5, 2007
(old Question 2 Postponed until next week)
(Maple worksheet postponed for a while.)
  • Read Sections 1.4 and 6.2.
  • Turn in well-written solutions to the following problems.
  • If you work with other students in the class, make sure to include a "Thank you" at the top of your homework assignment.
    • Question 1: 1.4.4abc
    • Question 2: (a) 6.1.1 (b) 6.1.2 [Don't forget to formulate the models, as it says in the problems. You may use eigenvalue and eigenvector calculations if you wish; make sure you explain what you're doing.]
    • Question 2: Consider two events E1 and E2 that we don't know whether or not they are independent.
      Will P(E1 AND E2) > P(E1)P(E2) or will P(E1 AND E2) < P(E1)P(E2)? Determine all that you can about this situation.
      [You may wish to consider multiple cases, perhaps investigating using Venn diagrams.]
    • Question 3: (Discussion) While brainstorming Example 1.4.1 in class, we made an (incomplete) list of information that we would need to consider when modeling the distribution of cars in Orlando and Tampa. We also made the assumption that all cars are rented out each day regardless of location. (a) Give a sentence or two of justification for this assumption. (b) Cite two pieces of information (at least one of which was not on the board on Friday) that you would ask for from the company in order to model the distribution of cars, and write at least one paragraph for each explaining how this additional data would change or have an impact on the model we discussed.
    • Question 4: [worth 4 pts] Project 6.2.1 on p226 [Important: this is NOT Problem 6.2.1]
    • Question 5: [worth 5 pts] (Maple) Complete and turn in your work from Thursday's lab section.
Homework 2
due Monday, January 29, 2007
  • Read the syllabus. This should answer any questions you may have about the structure of the class.
  • Read Sections 1.0, 1.1, 1.2, and 1.3.
  • Complete the following problems. Your answer sheet should be neat and well organized. AND, all answers should be complete with explanations.
    • Question 1: (Multiple Computations) 1.1.3bcd, 1.3.1cde [Revised 1.1.3c: {2, 5, 11, 23, 47, 95, 191}]
    • Question 2: (Data Collection & Plotting) 1.3.3ab (You should generate at least 30 data points in each.)
      [Typo alert: in (a), an n is missing: it should be an+1=--1.2an+50]
    • Question 3: (Word Problem) (a) 1.2.6 (b) continued in 1.3.11 (you may also wish to consult 1.2.7)
    • Question 4: (Word Problem) 1.3.7
    • Question 5: (Open ended paragraphs) (a) 1.1.6 (name two not described in the book, and explain why you expect them to have this behavior) ) (b) Give an example of a behavior that would take a higher order recurrence relation. That is, a dynamical system where an+1=c*an+d*an-1, for non-zero constants c and d. Don't forget to explain why.
  • Note: This is Homework 2 because it's due on Week Number 2. Just a little quirk. Deal! :)

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Binghamton University  Department of Mathematical Sciences