Please spend at least 2 hours a night reading the material/looking at the proofs/making sure you understand the details. Below is a tentative reading list and homework assignments. It is subject to changes depending on the amount of material covered each week. I strongly encourage you to skim the reading before class, so you are familiar with the definitions, concepts, and the statements of the material we'll cover that day.

• Week 13: Nov 24, 2014:

  • Work  on exam (first part due by 10:30am on Wednesday) and projects.

• Week 12: Nov 17 to 21, 2014:

  • Read my paper with colleagues on the movie industry: http://web.williams.edu/Mathematics/sjmiller/public_html/math/papers/SilverScheduler_finalver_05012007.pdf (especially read section 4 on demand forecasting). Some press on the paper: http://www.emac-online.org/userfiles/file/EMAC%20Newsletter_2nd_2010_last.pdf (page 9).

  • We will have talks by myself and students on programming.

• Week 11: Nov 10 to 14, 2014:

  • Keep working on your projects. Read Section 3.4. Rental harmony paper: http://www.maa.org/sites/default/files/pdf/upload_library/22/Hasse/00029890.di011943.01p0581t.pdf

  • HW: Due Monday November 17: Textbook: Page 271: #1. Also: Consider the map from the unit circle (all points \((x,y): x^2 + y^2 \le 1\) to itself given by \(f(x,y) = ((y-1/2)^2/8, (x-1/2)^2/8)\). Does this map have any fixed points? Why or why not. If yes find or approximate it.

  • HW: Optional: Page 271: #5, #6, #14.

• Week 10: Nov 3 to 7, 2014:

  • Keep working on your projects. Work on exam.

  • Reading: If you want to read about contraction maps and difference equations: http://web.williams.edu/Mathematics/sjmiller/public_html/209/handouts/firstorderexistunique.pdf

• Week 9: Oct 27 to 31, 2014:

  • Keep working on your projects. Prepare for exam: will get towards end of week. (No written HW: the HW will now be due on Monday November 3rd, though I urge you in the strongest possible way to have it done and hand it in on Friday).

    • Homework: Due Monday November 3: Section 2.3.1 of my notes: Exercise 2.7 (The notes might not have been clear: take as the original problem \(A^T x \le b\), \(x\) arbitrary, minimize \(c^T x\), and take the dual problem to be \(y^T A \ge c^T\), \(y\) arbitrary, minimize \(y^T (-b)\)). Problem \#2: Medical Residencies: Imagine there are \(P\) people who have just graduated from medical school and \(H\) hospitals. We are trying to match medical students with hospitals. Each student ranks the hospitals and each hospital ranks the students. Formulate this assignment problem as a linear programming problem; you may need to make some assumptions to finish the modeling. There are a lot of ways to do this; what do you want to maximize? Does a feasible solution always exist, and if so when? Does the existence of a feasible solution depend on the function you want to optimize? Problem \#3: Exercise 2.10 from the notes. Note this is the \(A'\) from the text, and thus the \(k\) columns of \(A'\) are linearly independent. \#4: Exercise 2.11 from the notes. \#5: Write down linear constraints for the event \(A\) or \(B\) or \(C\) must happen. \#6: Consider an \(n \times n \times n\) chesscube. Write down a linear programming problem to figure out how many hyperpawns can safely be placed given that \(n\) hyperqueens are placed in the chesscube. Note the hyperqueens can attack diagonally, horizontally, vertically, and forward-backly.

  • Exam: I will give out the exam on Friday and it will be due late on November 8th or sometime on November 9th.

  • I am modifying when the HW and when the exam is due to accomodate requests to adjust the due dates for spirit week; as mentioned above, however, I strongly urge you to do the work when you can so it doesn't pile up, but I will give you the opportunity to make your choices.

  • Class on Monday at WCMA -- please email me if you are coming (if at least 5-10 are not I will cancel the WCMA trip and have a regular class). We will look at some real world optimization problems, from the log rule to lights and furniture. Supplemental Lecture: Simplex Method Example: Part I (http://youtu.be/v0tzbA0vwxM) and Part II (http://youtu.be/CvyOpydExKc). Also view https://www.youtube.com/watch?v=bYIYuWbFSg0&feature=youtu.be for an example of when an integer max is near the real max (start at 25:30 and go to the end). The problem is: Given a positive integer \(S\), write it as a sum of positive integers \(a_i\) such that the product of the \(a_i\)'s is as large as possible. There  are a lot of good techniques to attack problems like this. If you are interested in problems like this, the section immediately before this problem might also be of interest to you, where we show \(e^\pi > \pi^e\) by elementary means (lot of Taylor series, but not using a calculator!). This is a great way to see how to use calculus knowledge.... That starts at 11:35.

  • Skim the chapter in the book on fixed point theorems. Read Chapter 3, Section 1 of the book, and then skip ahead and read Chapter 3, Section 4 of  the textbook.

  • Keep working on your projects: click here for a list of who is doing what (if you're not listed let me know). Every few days shoot me a progress report.

• Week 8: Oct 20 to 24, 2014:

  • Keep working on your projects

  • Reading: TBD, including rest of Chapter 3 of notes, Section 1.13 of the book (page 104, just skim).

  • Homework: Due Friday October 31 (changed to Monday November 3): Section 2.3.1 of my notes: Exercise 2.7 (The notes might not have been clear: take as the original problem \(A^T x \le b\), \(x\) arbitrary, minimize \(c^T x\), and take the dual problem to be \(y^T A \ge c^T\), \(y\) arbitrary, minimize \(y^T (-b)\)). Problem \#2: Medical Residencies: Imagine there are \(P\) people who have just graduated from medical school and \(H\) hospitals. We are trying to match medical students with hospitals. Each student ranks the hospitals and each hospital ranks the students. Formulate this assignment problem as a linear programming problem; you may need to make some assumptions to finish the modeling. There are a lot of ways to do this; what do you want to maximize? Does a feasible solution always exist, and if so when? Does the existence of a feasible solution depend on the function you want to optimize? Problem \#3: Exercise 2.10 from the notes. Note this is the \(A'\) from the text, and thus the \(k\) columns of \(A'\) are linearly independent. \#4: Exercise 2.11 from the notes. \#5: Write down linear constraints for the event \(A\) or \(B\) or \(C\) must happen. \#6: Consider an \(n \times n \times n\) chesscube. Write down a linear programming problem to figure out how many hyperpawns can safely be placed given that \(n\) hyperqueens are placed in the chesscube. Note the hyperqueens can attack diagonally, horizontally, vertically, and forward-backly.

• Week 7: Oct 13 to 15, 2014:

  • Finish reading Chapter 3 of my notes

  • Homework: Due Friday Oct 17: #1: Formulate Sudoku as a linear programming problem (you can do either \(4\times 4\) or \(9\times 9\) Sudoku). #2: Consider the \(3 \times 3\) constraint matrix \(A\) where the first row is 1, 2, 3, the second row is 4, 5, 6 and the third row 7, 8, 9 (thus it's the numbers 1 through \(3^2\)). Let the vector b equal \((1, 1, 1)^T\). Find all basic feasible solutions to \(A x = b\) with \(x \ge 0\).  #3: Let's revisit the chess problem from class. Consider an \(n \times n\) chess board. We want to put down \(n\) queens and maximize the number of pawns that can be safely placed on the board. Set this up as a linear programming problem. #4: Do Exercise 2.14 from my notes. #5: Hand in a short write-up saying who is in your group and what you will be studying / doing. Give a brief outline of what you think you'll need to learn, what data you think you'll need to gather, .... Describe why you feel your group has the necessary skill sets to complete the task, or if not what your plan is to remedy that.

• Week 6: Oct 6 to Oct 10, 2014:

  • Finish reading Chapter 3 of my notes.

  • Homework: Have your first Team Meeting by Wednesday, October 15.

  • Homework: Due Friday Oct 17: #1: Formulate Sudoku as a linear programming problem (you can do either \(4\times 4\) or \(9\times 9\) Sudoku). #2: Consider the \(3 \times 3\) constraint matrix \(A\) where the first row is 1, 2, 3, the second row is 4, 5, 6 and the third row 7, 8, 9 (thus it's the numbers 1 through \(3^2\)). Let the vector b equal \((1, 1, 1)^T\). Find all basic feasible solutions to \(A x = b\) with \(x \ge 0\).  #3: Let's revisit the chess problem from class. Consider an \(n \times n\) chess board. We want to put down \(n\) queens and maximize the number of pawns that can be safely placed on the board. Set this up as a linear programming problem. #4: Do Exercise 2.14 from my notes. #5: Hand in a short write-up saying who is in your group and what you will be studying / doing. Give a brief outline of what you think you'll need to learn, what data you think you'll need to gather, .... Describe why you feel your group has the necessary skill sets to complete the task, or if not what your plan is to remedy that.

• Week 5: Sept 29 to Oct 3, 2014:

  • Finish reading chapter 2 of my notes.

  • Homework Problems: Due Friday, Oct 3: Exercise 2.10, Exercise 2.11.  Also: #3: Imagine you want to transmit the shape of the plot \(f(x) = \sin(x^3)\) on the interval [-3,3]. You have the ability to sample the value of this function for 360 different choices of x. Plot it if you sample uniformly. Is this the best way to sample? How should you sample / choose where to sample? #4: We say a x is an ordered feasible solution if its non-negative entries are ordered from smallest to largest; thus (1,0,0,4,3,0,0,5,8) is not ordered (as 4 is less than 3) but (1,0,0,3,3,0,0,5,8) is. Prove or disprove: if a canonical linear programming problem has a feasible solution then it has an ordered feasible solution. #5: Give an example of a 4x4 matrix such that each entry is positive and all four columns are linearly independent; if you cannot find such a matrix prove that one exists. #6: Redo the previous problem but for an arbitrary N (thus find an NxN matrix where all entries are positive and the N columns are linearly independent). Extra Credit: For each positive integer N find a matrix with N rows and infinitely many columns so that all entries are positive and any set of N columns is linearly independent.

  • Homework: Start forming a team to do a project for the class! Think about what kind of project you want to work on (theoretical, applied), and what skill sets you will need.

• Week 4: Sept 22 to 26, 2014:

  • Read (in my notes) through Section 2.4, and the corresponding sections of the book.

  • Homework problems: Due Friday, September 26: #1: Investigate the Euclidean algorithm for various choices of \(x\) and \(y\). What values cause it to take a long time? A short time? For problems like this you  need to figure out what is the right metric to measure success. For example, if \(x < y\) and it takes \(s\) steps, a good measure might be \(s / \log_2(x)\).  Section 2.2.3 of my notes: Exercises 2.3, 2.4, 2.5. Final problem: the diet problem with two products and two constraints led us to an infinite region, and then searching for the cheapest diet led us to a vertex point. Modify the diet problem by adding additional constraints so that, in general, we have a region of finite volume, and again show that the optimal point is at a vertex. Your constraints should be reasonable, and you should justify their inclusion. 

  • You should know LaTeX and Mathematica (or another programming language): tutorials are here: http://web.williams.edu/Mathematics/sjmiller/public_html/math/handouts/latex.htm

• Week 3: Sept 15 to 19, 2014:

  • Homework problems: Due Friday, September 19 #1: Write a program to generate Pascal's triangle modulo 2. How far can you go? Can you use the symmetries to compute it quickly? You do not need to hand this problem in. #2: What is the dimension of the Cantor set? #3: There are many trig tables online (see for example http://www.sosmath.com/tables/trigtable/trigtable.html). Using the look-up table, discuss a simple way to estimate the value of the sine of any angle, then discuss a generalization that will be more accurate. For your better interpolation, provide a bound on your error as a  function of the angle. #4: Redo the previous question, but now for interpolating values of the standard normal (http://www.sjsu.edu/faculty/gerstman/EpiInfo/z-table.htm). #5: In class we got a lower bound for \(1 + \cdots + n\) by saying it was at least \(n/2 \ast n/2\). Instead of breaking it at the middle, if we consider \(cn, cn+1, \dots, n\) we get a contribution of \(cn \ast (1-c)n\). Show this is largest when \(c = 1/2\).

  • You should know LaTeX and Mathematica (or another programming language): tutorials are here: http://web.williams.edu/Mathematics/sjmiller/public_html/math/handouts/latex.htm

• Week 2: Sept 8 to 12, 2014:

  • Review your linear algebra (see https://www.youtube.com/watch?v=f7ii-sjfBrA&feature=youtu.be and http://youtu.be/yGVq-zx6Whg).

  • Read ahead in the course book and my notes -- I'll be lecturing on a hodgepodge of topics you have hopefully seen before, but casting them in a new light. Use this time to make sure your linear algebra is strong and build a strategic reserve of material read for later.

  • Homework problems: Due Friday, September 19 #1: Write a program to generate Pascal's triangle modulo 2. How far can you go? Can you use the symmetries to compute it quickly? #2: What is the dimension of the Cantor set? #3: There are many trig tables online (see for example http://www.sosmath.com/tables/trigtable/trigtable.html). Using the look-up table, discuss a simple way to estimate the value of the sine of any angle, then discuss a generalization that will be more accurate. For your better interpolation, provide a bound on your error as a  function of the angle. #4: Redo the previous question, but now for interpolating values of the standard normal (http://www.sjsu.edu/faculty/gerstman/EpiInfo/z-table.htm). #5: In class we got a lower bound for \(1 + \cdots + n\) by saying it was at least \(n/2 \ast n/2\). Instead of breaking it at the middle, if we consider \(cn, cn+1, \dots, n\) we get a contribution of \(cn \ast (1-c)n\). Show this is largest when \(c = 1/2\).

• Week 1: Sept 1 to Sept 5, 2014:

  • Welcome slides for first class.
  • Handout for first class.
  • Review your linear algebra: review lectures online at thttp://web.williams.edu/Mathematics/sjmiller/public_html/linprog/videos/LinProg_Lect01_LinAlgReview.MP4 and http://web.williams.edu/Mathematics/sjmiller/public_html/linprog/videos/LinProg_Lect02_LinAlgReview.MP4
  • Start exploring mathematical software (R, Mathematica). Learn LaTeX. I have handouts and videos: http://web.williams.edu/Mathematics/sjmiller/public_html/math/handouts/latex.htm
  • Homework: To be emailed to me by 8pm on Sunday, September 7: Email me a short note (a paragraph suffices) on what you want to get out of this course, and what lesson  you learned from the graduation speech by Uslan (http://www.graduationwisdom.com/speeches/0018-uslan.htm). Full credit, 20/20, so long as you answer both questions on time.
  • Homework: To be done by start of class on Monday, September 8: Read Section 1 of the book (to page 11).
  • Homework: To be done by start of class on Friday, September 12: find something in your life that you can do better / more efficiently, and tell me about it (in person or email). You should constantly be thinking about how to optimize (but please think about what you choose to share!).