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: November 28 to December 2, 2016: 
  • Read: Walmart articles (see additional comments)
  • HW: THINK about Voronoi cell construction for Wednesday: try to beat order \(n^2 \log n\) and get something roughly of order \(n\).
  • Keep working on chapter, project, letter of rec
   

• Week 12: November 21, 2016:
  • Keep working on chapters / projects

• Week 11: November 14 to 18, 2016:
  • Read: Chapter 16, http://www.maa.org/sites/default/files/pdf/upload_library/22/Hasse/00029890.di011943.01p0581t.pdf
  • Homework: Hand in first draft of letter of recommendation, hand in outline of project, hand in outline of chapter work.

   

• Week 10: Nov 7 to 11, 2016:

  • Read: Chapters 15 and 16 

  • Homework: Due November 14: \#1. In the plane we have \(\overrightarrow{v}^{(0)} = (3,-2)\), \(\overrightarrow{v}^{(1)}= (1,5)\) and \(\overrightarrow{v}^{(2)} = (-7,1)\). Let \(\overrightarrow{x}\) have cartesian coordinates \((c_1,c_2)\) and barycentric coordinates \((x_0,x_1,x_2)\). Write the cartesian coordinates in terms of the barycentric coordinates, and vice versa. \#2: 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. \#3: Consider \(x_n = \cos(n)\) (measured in radians). The Bolzano-Weierstrass Theorem asserts it has a subsequence which converges to a point in \([-1, 1]\). Explicitly find such a subsequence. Is it easier, harder or the same to prove the analogous statement for \(y_n = \sin(n)\)? Do so. Hint: you may use properties of \(\pi\), such as its decimal or continued fraction expansion. Also, if you have a homework exemption could be a good time to use it....

   

• Week 9: Oct 31 to Nov 4, 2016:

  • Read: Chapters 15 and 16

   

• Week 8: Oct 24 to 28, 2016:

  • Keep working on your projects.

  • Make sure you have chosen a chapter of the book, confirmed it with me, and start working on that.

  • Choose a person in the class to write a letter of recommendation for, and ask them. Everyone must write at least one letter, if you wish to do more you may. Confirm with me when you agree to write a letter for someone. You are not writing the letter now, just finding people. If no one asks you to write a letter I will assign someone to you.

  • Homework: Due Monday October 31: Problem \#1: Exercise 8.7.18. Frequently in problems we desire two distinct tuples, say points \((a_1, \dots, a_k) \neq (\alpha_1, \dots, \alpha_k)\). Find a way to incorporate such a condition within the confines of integer linear programming. 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 9.3.5. Modify the decomposition problem so that we write S as a sum of non-negative summands, but now we want to maximize the product of the squares of the summands; what is the answer?  \#4: Exercise 9.3.13. Generalize the knapsack problem so that in addition to needing the total weight to be below a critical threshold, there is also a volume constraint. Set this up as a linear programming problem.  \#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 17 to Oct 24, 2016:

  • Read  Chapters 13 and 14

  • Homework: Due Monday Oct 24: No written assignment. Continue making progress on your group project, and start working on your book chapter.

   

• Week 6: Oct 10 to Oct 14, 2016:

  • Read  Chapters 10,  11 and 12

  • Homework: Due Monday 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 6.6.30. \#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, what you've done so far .... 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: Oct 3 to 7, 2016:
  • Read: Chapters 7, 8 and 9.
  • Homework due Monday October 10: \#1. Prove that if \(A'\) has \(M\) rows and \(k\) columns, with \(M \ge k\), then \(A'^T A'\) is invertible. Note this is the \(A'\) from the text, and thus the \(k\) columns of \(A'\) are linearly independent.\#2. For fixed \(M\), find some lower bounds for the size of \(\sum_{k=1}^M \left({N \atop k}\right)\). If \(M = N = 1000\) (which can easily happen for real world problems), how many basic feasible solutions could there be? There are less than \(10^{90}\) sub-atomic objects in the universal (quarks, photons, et cetera). Assume each such object is a supercomputer capable of checking \(10^{20}\) basic solutions a second (this is much faster than current technology!). How many years would be required to check all the basic solutions? \#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 \(4\times 4\) 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 \(N\times N\) 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.

   

• Week 4: Sept 26 to 30, 2016:
  • Read: Chapters 4, 5 and 6 of my textbook.
  • Homework due Monday October 3: \#0: Make sure you can do, but do not submit, the problems in the ``Real Analysis Review'' of Chapter 4. \#1: Exercise 4.7.7: Prove that any cubic \(ax^3 + bx^2 + cx + d = 0\) can be written as \(x^3 + p x + q = 0\) (i.e., we can rewrite so that the coefficient of the \(x^2\) term vanishes and the coefficient of the \(x^3\) term is 1); this is called the depressed cubic associated to the original one. (For fun see the next problem on how to solve the cubic.) \#2: Exercise 4.7.12. Can you construct a canonical linear programming problem that has exactly two feasible solutions? Exactly three? Exactly \(k\) where \(k\) is a fixed integer? \#3: Exercise 4.7.14. Find a continuous function defined in the region \((x/2)^2 + (y/3)^2 < 1\) (i.e., the interior of an ellipse) that has neither a maximum nor a minimum but is bounded. \#4: Exercise 5.4.3. Imagine we want to place \(n\) queens on an \(n \times n\) board in such a way as to maximize the number of pawns which can safely be placed. Find the largest number of pawns for \(n \le 5\). \#5: Exercise 5.4.4. Write a computer program to expand your result in the previous problem to as large of an n as you can. Does the resulting sequence have any interesting problems? Try inputting it in the OEIS. \#6: Consider the problem of placing \(n\) queens on an \(n\times n\) board with the goal of maximizing the number of pawns which may safely be placed. For each \(n\), let that maximum number be \(p(n)\). Find the best upper and lower bounds you can for \(p(n)\). For example, trivially one has \(0 \le p(n) \le n^2\); can you do better? \#7: Exercise 5.4.26: Prove \(\frac{1}{\sqrt{N}} \sum_{n=-\infty}^\infty e^{-\pi n^2/N} \ = \ \sum_{n=-\infty}^\infty e^{-\pi n^2 N}.\) As \(N\) tends to infinity, bound the error in replacing the sum on the right hand side with the zeroth term (i.e., taking just \(n=0\)). Hint: the Fourier transform of a Gaussian is another Gaussian; if \(f(x) = e^{-ax^2}\) then \(\widehat{f}(y) = \sqrt{\pi/a} e^{-\pi^2 y^2/a}\).

• Week 3: Sept 19 to 23, 2016:
  • 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 lecture 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 Monday, September 26: \#0: Choose a project; put that on the google sheet: https://docs.google.com/spreadsheets/d/178i44ycRXjfVsHQZSpgngrWMZvdJgH9gd1CSyAXpMRg/edit#gid=0. \#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 \(s / \log_2(x)\).  \#2: What is the dimension of the Cantor set? \#3: Exercise 3.6.38: Find the optimal solution to the diet problem when the cost function is \({\rm Cost}(x_1,x_2) = x_1 + x_2\). \#4: Exercise 3.6.39. There are three vertices on the boundary of the polygon (of feasible solutions); we have seen two choices of cost functions that lead to two of the three points being optimal solutions; find a linear cost function which has the third vertex as an optimal solution. \#5: Exercise 3.6.40. Generalize the diet problem to the case when there are three or four types of food, and each food contains one of three items a person needs daily to live (for example, calcium, iron, and protein). The region of feasible solutions will now be a subset of \(\mathbb{R}^3\). Show that an optimal solution is again a point on the boundary. \#6: 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. 

• Week 2: Sept 12 to 16, 2016:
  • 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 lecture 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 Monday, September 19: #0: 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 in. From the textbook: #1: Exercise 1.7.4 (there are many trig tables online: see for example http://www.sosmath.com/tables/trigtable/trigtable.html), and read BUT DO NOT DO Exercise 1.7.5. #2: Exercise 1.7.7. #3: Exercise 1.7.18. #4: Exercise 1.7.34. #5: Exercise 1.7.26. #5: Exercise 1.7.36.

• Week 1: Sept 9, 2016
  • 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 noon on Sunday, September 11: 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). Also send me a resume/CV (send as a pdf, with a good filename -- CV.pdf or resume.pdf is a sure way to make me angry).
  • Homework: To be done by start of class on Monday, September 12: Read Chapter 1.
  • Homework: To be done by start of class on Friday, September 16: 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!).