**Homework
Assignments**

** **

The homeworks are modeling problems. As such, they rarely if
ever have correct or incorrect answers. All that is asked is that you __explore__
the topics assigned in teams of 2-3 students and write a reasoned answer in
good technical English. All members of the team must read and edit the text of
your solution before it is turned in. Aim for clear concise prose.

The homeworks can be turned in via BlackboardŐs Digital Dropbox feature (preferred) or emailed to salamon@sdsu.edu. I am hoping BlackboardŐs Discussion feature will also prove useful in enabling the discussions / revisions needed to work with the other members of your team.

** **

**Week 1**: Read chapter
1 of BenderŐs book. Do problems 1, 2, 7, and 8 at the end of the chapter (pp.
12-14).

**Due** date September 10

**Week 2**: Read section
2.1 of BenderŐs book. Do problems 4, 5, and 6 at the end of the section (pp.
31-34).

**Optional assignment**:
It is hoped that each of the final projects during the last third of the
semester come from a real modeling problem related to a
thesis/dissertation/research project by some member of the class. If you have
such a problem and would like to work on it as part of your efforts for this
class, please submit a few sentences describing your problem.

**Due** date September 17

**Week 3**: Read section
2.2 of BenderŐs book. Do problems 1, 2, and 3 at the end of the section (pp.
40-42).

Starting with this homework set, you may substitute a
summary paragraph on the weekŐs computational science seminar or a paragraph
written to one of the online threads for __one__ of the problems (your
choice).

**Due** date September 24

**Week 4**: Read
sections 3.1 and 3.2 of BenderŐs book. Do problems 6, 7, and 8 at the end of
the section (p. 57).

**Due** date October 1

**Week 5:** Read section
3.3 of BenderŐs book. Do problems 1, 3, and 4 at the end of the section (p.
63-4).

**Due** date October 8

**Week 6:** Midterm 1

**Due** date October 15

**Week 7:** Read section
4.1 of BenderŐs book. Do problems 1, 2, and 3a-e at the end of the section (p.
76-7).

**Due** date October 29

**Week 8**: Read section
4.2 of BenderŐs book. Do problems 2, 3a-b, and the following game theory
problem:

GT1.
Consider the following game intended to be a simplified model of
bluffing in poker. It is played with a deck consisting of two cards: a high
card and a low card. After an ante of $1 by each player, the first player (P_{1})
is dealt one of the cards, each with equal probability. While P_{1} can
look at the card immediately, P_{2} is not allowed to see the card
until the end of the hand. The rules of the game are:

If
P_{1} gets the high card he must bet $1 while if he gets the low card
he can either bet $1 or he may fold, i.e. accept a loss of his $1 ante.

P_{2}
can respond to a bet of $1 either by calling P_{1}, and risking a total
of $2, or by accepting a loss of his $1 ante. If P_{1} bet and P_{2}
calls him then P_{1} wins or loses $2 according to whether he does or
does not hold the high card.

Formulate "pure" strategies of each player
and construct the payoff matrix. Note that this is a game with a chance move
(the dealing of the card) and so a_{11} is the __mean__ payoff to P_{1}
if both players use their respective first strategies many times. What is the
value of the game? What are the optimal mixed strategies of the two players?

**Due** date November 5

**Week 9**: Do problem
4a-d on page 99 and problem 5 on page 133 in BenderŐs book along with the
following maximum entropy problem:

ME1. The distribution of energies in a physical system maximizes entropy. A cavity contains 1000 electrons whose total energy is 750 ev. If the energy of each electron is quantized to be either 0, 1, or 2 ev., how many electrons have energy 0? If the temperature is the reciprocal of the Lagrange multiplier corresponding to the energy constraint, what is the temperature of the system?

**Due** date November 12

**Week 10**: Midterm 2

**Due** date November 19