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Week
13: November 26 to 30
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Week
12: November 19 to 23 (no class Wednesday or Friday)
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Week
11: November 12 to 16, 2012 (no class Monday Nov 12)
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Week
10: November 5 to 9, 2012
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Week
9: October 29 to November 2, 2012
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Keep working on your project.
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Keep reading random matrix theory. Two good introductions are
here and
here. Skimming is fine!
The slides are online here (click on the classical random matrix
theory section).
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Exam will be take-home and due around November 12th (for those in my 341
you may move it back a week as I know you have an exam there as well).
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Homework due Monday, November 5: #1: Let A = \mattwo{1}{2}{2}{1} (this is
the matrix with first row 1 2 and second row 2 1). Find the eigenvalues
and eigenvectors of unit length of A. If
\overrightarrow{v_1} and \overrightarrow{v_2} are these eigenvectors, let
Q be the matrix where the first column is \overrightarrow{v_1} and the
second column is \overrightarrow{v_2}. Compute Q^T A Q. #2: Let T_{n+1} =
T_n + T_{n-1} + T_{n-2} with T_0 = 0, T_1 = 0 and T_2 = 1. Find the
generating function for this sequence.
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Week
8: October 22 to October 26, 2012 (no class on Monday the 21st: use the time
for your projects)
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Keep working on your project.
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Keep reading random matrix theory. Two good introductions are
here and
here. Skimming is fine!
The slides are online here (click on the classical random matrix
theory section).
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Due Monday October 29: #1: Give an example of a square matrix A such that
there is no orthogonal matrix Q with Q^T A Q a diagonal matrix. #2: Let Q
be an orthogonal matrix. Must Q^3 be orthogonal? What about Q + Q + Q?
Prove your claims. #3: Consider NxN real symmetric matrices such that
each matrix element is at most B. Find as good as you can upper bound for
the absolute values of the eigenvalues in terms of B and N. #4: A unitary
matrix U is such that U^H U = U U^H = I, where H stands for the Hermitian
of the matrix (this means taking the complex conjugate of the transpose).
In class we proved the eigenvalues of real symmetric and complex
Hermitian matrices are real. Discover and prove as much as you can about
the eigenvalues of unitary matrices. What can you say about them? What
about the eigenvalues of orthogonal matrices?
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Week
7: October 15 to October 19, 2012 (no class on Monday the 21st: use the time
for your projects)
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Read: Skim from Franklin's book: Chapter 1, Sections 13 and 15.
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For Friday: Start reading random matrix theory. Two good introductions
are here
and
here. Skimming is fine!
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Think about the following problem for Wednesday: Given a large number (like 100, but in
general N) write it as a sum of positive integers a_1, ..., a_n such that
the product a_1 * ... * a_n is as large as possible. Is this a linear
programming problem? Note the number of partitions n is not pre-assigned.
What is the optimal solution? What if the a_i's just have to be positive?
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Homework: Due Wednesday, October 23 (no class on Monday, use the time for
your project): submit the introduction to your report (complete with
bibliography). This should describe the problem in detail, put it in
context of other, similar problems, describe the theoretical issues,
mention what you'll get to but not have the technical
details.
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Bridge conventions
online here (and here is a
link to the John Wayne Bridge Convention being mentioned on a 'real'
site).
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Week
6: October 8 to October 12, 2012 (no class on Monday the 8th)
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Reading: You can read Chapter 4 of my notes to see about linearizing
polynomial constraints.
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Think about how to write a general integer random variable as a sum of
binary random variables. You may assume your variable is between -N and N
for some large N.
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Homework: Due Monday, October 15: #1: Write an introduction to your
problem / topic. Clearly state what you are going to tackle. At the very
least, enumerate what additional material you'll need to discuss your
topic that has not been covered in class. If there are items you want me
to lecture on, let me know. Summarize some of the literature. This \emph{must}
be in TeX. It should be at least 3 full pages (this includes the
bibliography, but not a title page!). This is meant to be the nucleus of
your write-up. Presentations will start as soon as people are ready. You
have anywhere from 20 minutes to 50 minutes. Take the time to do a good,
thorough job. Your write-up should also be complete. You may assume your
audience has the knowledge base of our class; anything that we haven't
covered must be explained (though not necessarily proved).
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Week
5: October 1 to October 5, 2012
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If you want, read the notes in the book about how to linearize non-linear
conditions, and read about the
Strassen
algorithm (see
also the Mathworld entry here, which I think is a bit more readable)
if you want. A big theme of next week is to extend the applicability of
Linear Programming. I may teach the Method of Least Squares, as that's a
wonderful application of linear algebra and, like linear programming, is
far more applicable then you might believe. This is one of my goals for
the course, to get you to think about how to convert new problems to ones
you can do.
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HW: Due Monday, October 8: #1: Submit to me (separate from the rest of
your homework) an outline for your paper topic and class presentation.
This should include a summary of what you want to discuss, what you want
the class to get out of your presentation / writeup, what sources you
believe you will use. #2: Write down linear constraints for the event A
or B or C must happen. #3: Find as good of a function f as you can such
that you can find infinitely many pairs of integer x < y with the
run-time of the Euclidean algorithm at least f(x). For example, what we
did in class shows you can't take f(x) = 4 log_2(x); can you take f(x) =
c log_2 x for some c < 1? #4: Consider an n x n x 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.
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Extra Credit: for a couple of values of n, figure out the maximum number
of pawns that can safely be placed on an n x n chessboard given that
there are n queens that must be placed. Is this sequence in the
OEIS?
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Week
4: Sept 24 to Sept 28, 2012
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Continuing reading, including the proof of the simplex method.
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HW: Due Monday, October 1: #1: Choose a tentative topic for your paper
and class presentation, and a tentative group (of between 2 and 4
people). Have one person from the group email me a few paragraphs listing
all group members, describing the project and stating what you want the
class to get out of your write-up and talk. If you are having trouble
coming up with topics, let me know. #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: Prove Mz = w has either 0,
1 or infinitely many solutions, and no other options can happen. #4:
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. #5: Do
Exercise 2.14 from my notes.
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Extra credit: Modify #2 so that we have an n \times n matrix with the
entries going from 1 to n^2, with n \ge 3. Let b = (1, 1, \dots,
1)^T. Find all basic feasible solultions.
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Week
3: Sept 17 to Sept 21, 2012 (no class on Monday)
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Continue to review your linear
algebra; review lectures online at
http://web.williams.edu/Mathematics/sjmiller/public_html/linprog/videos/LinProg_Lect01_LinAlgReview.MP4
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Read up to and including Section 2.4 of the
notes, and the corresponding parts of the book.
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HW: Due September 24, 2012:
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 <= b, x arbitrary, minimize c^T x, and
take the dual problem to be y^T A >= c^T, y arbitrary, minimize y^T
(-b)). Problem #2: Formulate sudoku as a linear programming problem (you
can do either 4x4 or 9x9 Sudoku). Problem #3: 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 #4:
Exercise 2.10 from the notes. Note this is the A' from the text, and
thus the k columns of A' are linearly independent. #5: Exercise 2.11
from the notes.
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Additional: everyone is responsible for reading and presenting a topic in
linear programming, in groups of 2 to 4. Start thinking about what topic
you would like to choose, and whom you would like to work with.
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Optional: prove that a contraction map iterates any point to a fixed
point.
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Week
2: Sept 10 to Sept 14, 2012 (Due on Monday, but may be handed in on Wednesday
b/c Monday is Rosh Hashana, no class)
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Week
1: Sept 3 to Sept 7, 2012