/ Math 36

HW Assignments

  • Daily Work: Just as in any other subject, developing proficiency in mathematical modeling requires sustained, consistent effort. Between in-class assignments and brief writing assignments. These problems will mostly be short written responses or computational in nature and provide opportunities for you to review that material from that day's lecture.
  • Essays: The main assessments in this course will be in the form of 2-3 page essays, one for each of the large topic segments in the class. These assignments will generally require you to perform an analysis using relevant techniques from the section and draw conclusions based on your experiments. The associated essay will describe your efforts, analysis, and conclusions (specific instructions will be presented for each assignment).
  • Quizzes: Each of the topic segments will conclude with a short quiz covering the computational methods covered. These quizzes will occur approximately once every two weeks.

Homework Problems

Week Date Sections Daily Work
Essays and Quizzes

1 9/11 1.1

  1. Download MatLab and install it on your computer. The Dartmouth installation guide and license can be found here.
  2. Write one paragraph describing your personal goals for this course. In particular, if there are specific types of data or social frameworks that would be particularly interesting to you, this would be a great place to describe them.
9/13 1.2 and MatLab

No Homework

9/14 Individual Student Meetings
9/15 1.3

Find population data for at least 5 years for a city that is important to you and plot the population against time. Write one sentence explaining why you chose this city, one sentence describing the data (units, time interval, source, etc.), and one sentence explaining which of the models that we looked at in class best matches your data or why it doesn't match any of the models.

2 9/18 1.3

  1. Read ``Models of Human Population Growth'' (linked on the resources page). Choose one of the arguments that he makes for the inadmissability of the logistic model and describe whether you think it is important or relevant.
  2. Using the logistic model MatLab code on the resources page, find a set of parameters (P_0,k,n,m) that displays chaotic behavior.
Due: 10/2

Goal: Write a 2-3 page essay describing a modification of the logistic equation for population growth.

In class we have discussed some of the properties of the model P_{n+1)=P_n+k(M-P_n)P_n as well as its predictions and limitations. Your task is to identify a single term that you think should be added to this difference equation and discuss the expected consequences of modifying the model. You should explain why you believe the term you have added is important, what type of data you would need to estimate the new effects, and how you believe the addition will make the model more accurate/useful.

Once you have settled on a modification, your writing should be motivated by the following question: ``How will the earth's population grow over time?'' In the essay, show your analysis and results and discuss their implications to this question. Be sure to discuss the strengths and limitations of the model and their impact on your confidence in the conclusions you draw.

To help you along the way, here are some questions that may help guide your analysis. Discussion of these need not appear in the essay you turn in but will likely help you perform a thorough analysis on which to base your writing.
  1. What are the meaning of the terms of the model? What assumptions to they imply? How are they interpreted?
  2. What are reasonable magnitudes for the constants? Why are values outside this range inappropriate?
  3. Can the population ever exceed the carrying capacity? If so, where are the boundary cases (i.e. what combinations of constants create one situation or the other?)?
  4. Under what conditions is the model appropriate/applicable? Are these conditions extant today? Do we expect them to continue in the future?
  5. How do your modifications affect the arguments made in ``Overpopulation is not the problem'' and ``Models of Human Population Growth''?
9/30 1.4

Read ``Overpopulation is not the problem'' (linked on the resources page). Write at most a paragraph comparing and constrasting this piece to Monday's reading. In particular, how do the viewpoints of the authors differ?

9/22 Chapter 2

Describe the modificiation to the logistic model that you are proposing for your writing assignment.

3 9/25 Chapter 3

Do problem 2 in Section 3.3 of your textbook (typed below). The numerical answers are in the back of your textbook and you should be able to check your answers in MatLab as well. Note that the textbook uses $m$ for the number of data points.

Use the normal equations $$a=\dfrac{m\left(\sum_{i=1}^mx_iy_i\right) - \left(\sum_{i=1}^n x_i\right)\left(\sum_{i=1}^n y_i\right)}{m\left(\sum_{i=1}^nx_i^2\right)-\left(\sum_{i=1}^mx_i\right)^2}$$ and $$b=\dfrac{\left(\sum_{i=1}^mx_i^2\right)\left(\sum_{i=1}^my_i\right)-\left(\sum_{i=1}^n x_iy_i\right)\left(\sum_{i=1}^n x_i\right)}{m\left(\sum_{i=1}^nx_i^2\right)-\left(\sum_{i=1}^mx_i\right)^2} $$ for the following data sets:

  1. $\{(1.0,3.6),(2.3,3.0),(3.7,3.2),(4.2,5.1),(6.1,5.3),(7.0,6.8)\}$
  2. $\{(29.1,.0493),(48.2,.0821),(72.7,.123),(92.0,.154),(118,.197),(140,.234),(165,.274),(199,.328) \}$
  3. $\{(2.5,4.32),(3.0,4.83),(3.5,5.27),(4.0,5.74),(4.5,6.26),(5.0,6.79),(5.5,7.23) \}$
9/27 Data Analysis Day

What are the equilibrium states for the following system: $$ a_{n+1}= a_n + 3a_n+4b_n -2 $$ $$ b_{n+1}=b_n+ 2a_n-3b_n +1$$

9/29 Preference Systems

Read an article on the ranked choice voting controversy in Maine (a couple are linked on the resources page). Think about how mathematical models could inform this debate (you don't have to hand anything in).

4 10/2 Lecture Notes

  1. Construct a preference graph on 5 options where each option has an equal number of wins and losses
  2. Explain why no such graph exists for 6 options.
  3. Construct a preference graph for 6 options where three options have three wins and three options have three losses.
10/4 Lecture Notes

Consider the preference system: $$\begin{array}{|c|c|c|c|} \hline 1&2&3&1\\ \hline \hline A&B&D&A\\ \hline B&C&A&C\\ \hline C&A&C&B\\ \hline D&D&B&D\\ \hline \end{array}$$ Determine the winners under the plurality, Borda, and Condorcet methods? Which option do you think should win a fair election?

Due: 10/16

Goal: Write a 2-3 page essay describing a game theoretic model of a local controversy. Details The purpose of this assignment is to get some experience with the considerations that go into formulating game theory models. We will look at many examples in class where the essential details have been nicely expressed in such a way that the game provides a desired conclusion using fairly simple analytical techniques. However, people are complicated, and there are often many different, equally valid ways to balance the utility calculations. You should select a controversy that has been covered by {\em The Dartmouth} at least once in the past two years and formulate a game theoretic model for the conflict. You should begin by identifying the competitors, describing their respective options, and formulating a decision matrix representing the conflict. Your essay should justify these choices, particularly the relative values of the entries in the decision matrix. After formulating your model you should express your hypotheses about the behavior of the model and its relevance to the actual outcome of the controversy. Then, perform an analysis using the tools that we have discussed in class for 2 player (non--cooperative) games to determine the likely results of the game from a mathematical perspective. Discuss your results in terms of your expectations, does the analysis agree with your intuition? Why or why not? After performing the analysis and considering the realistic outcome, how might you have modified the original model? To help you along the way, here are some questions that may help guide your analysis. Discussion of these need not appear in the essay you turn in but will likely help you perform a thorough analysis on which to base your writing.

  1. What are the meaning of the specific matrix entries? What assumptions do they imply? How are they interpreted in this specific case?
  2. How do the differences in magnitudes between the options of a single player affect the analysis? What about the differences between the values available to the two players?
  3. Does your game match in structure or outcome one of the stylized models we analyzed in class? What are the differences between your model and that one?
  4. How does our discussion of preference rankings and voting systems affect your analysis? What models would make the most sense if the members of one or both of the parties had to rank the options and vote for the strategy to select?

10/6 Lecture Notes

Assume that the preference system: $$\begin{array}{|c|c|c|} \hline 4&3&5\\ \hline B&A&C\\ \hline A&B&A\\ \hline C&C&B\\ \hline\end{array} $$ is known to all parties and that candidates $A$ and $B$ are allowed to play a game to determine the final voting method, choosing between options plurality, Borda (with 10 points for first, 2 for second, 1 for third), or Condorcet. Each candidate would prefer to win the election and $A$ and $B$ would both prefer that $C$ win rather than the other player. If the candidates select the same method that method is chosen, otherwise one of the three methods is selected uniformly at random. Write a paragraph describing which method you believe each of $A$ and $B$ should select and justifying your selection. Does one of the players have more leverage than the other? Why or why not?

5 10/9 Lecture Notes

  1. Construct a normal form matrix where player 1 has a dominated strategy but player 2 does not.
  2. Determine the values in a normal form matrix for the game consisting of two '21s running around the homecoming fire. Their options are to keep running laps like normal or try to touch the fire. Is your formulation a dilemma?

10/11 Lecture Notes

  1. Use IESDS to find reduce the following game: $$\begin{array}{|c|c|c|} \hline 6,8&2,6&8,2\\ \hline 8,2&4,4&9,5\\ \hline 8,10&4,6&6,7\\ \hline \end{array}$$
  2. What is the Nash equilibrium for: $$\begin{array}{|c|c|c|} \hline 4,3&5,1&6,2\\ \hline 2,1&8,4&3,6\\ \hline 3,0&9,6&2,8\\ \hline \end{array}$$

10/13 Lecture Notes

Input some text you have written (perhaps an essay for this class) to the Markov text generator here: link and look at the output. Would the output of the computer program get an A in this class? (you don't have to hand anything in)

This is not required, but if you are interested in the practical aspects of working with data, the following description of the process to construct a data set for generating Markov comics is an interesting read. Notice how much effort is required to format the data in a useful fashion before any actual mathematics can be applied. This is becoming an increasingly important feature of ``data-drive'' jobs, that is rarely discussed in the classroom.

6 10/16 6.1
In Hanover, we have four types of weather, sunny, cloudy, snowy, and rainy. If it rains or snows then the next day has a 40% chance of the other type of precipitation, a 30% chance of the same type of precipitation, a 20% chance of clouds, and a 10% chance of sun. When it is sunny, the next day is sunny 65% of the time, cloudy 25% of the time, and each type of precipitation happens 5% of the time. Finally, when it is cloudy, the next day any of the options could happen with equal probability. Formulate a Markov chain for this situation and determine the associated steady state. What is the probability that the weather for this schoolweek is exactly, sunny, cloudy, cloudy, snowy, sunny if it rained on Sunday? What is the most likely weather on Saturday if it snows on Thursday?
10/18 Lecture Notes
Modify the chain from Monday's assignment under the ``winter assumption'' that if it snows one day then it will continue to snow, every day, forever :( If it is sunny today, how many days is it expected to take until it begins snowing and how many of those days are likely to be cloudy?

Due: 10/30

Goal: Write a 2-3 page essay analyzing the use of Markov chains for text analysis.

Data and Code: Download link

Details: The purpose of this assignment is to explore the use of Markov models and random processes. In class, we looked at some simple examples of generating text using different order Markov models but in this assignment we will also be analyzing the behavior as the underlying text changes. I have provided transition matrices and labels for 1--4 order Markov chains for several input texts and the associated MatLab code to generate samples from each chain. There are two components to this assignment, one focusing on comparing the properties of specific authors and another on the sample outputs.

Both Markov and Shannon were interested in the individual transition probabilities between letters as a method for analyzing interesting features of specific authors. Using properties of the first order Markov chains, determine a criterion for attempting to distinguish between works of different authors and justify your decision. The goal is to formulate a rule that allows you to determine whether a particular piece of writing was authored by one of the authors in our set, or not. You may consider any aspects of the first order chain, including its steady state behavior, convergence time, or some of the specific matrix entries.

For the longer chains, the task is to analyze how increasing the order of the Markov chain affects the structure of the output strings. For each of the chains, you should generate several example texts of varying lengths and discuss your observations. Discuss how the features of the text, such as length, genre, or time period, relate to the outputs that you observe. Pay particular attention to the repetitions that occur as you generate longer text examples and discuss what aspects of the original text are responsible for them.

To help you along the way, here are some questions that may help guide your analysis. Discussion of these need not appear in the essay you turn in but will likely help you perform a thorough analysis on which to base your writing.

  1. Can you determine anything about the average length of the words from the first order chain? Are there any differences in vowel distribution? What types of letters are most likely to display useful anomalies for distinguishing between authors?
  2. How does the prevalence of particular words or proper nouns affect the observed behavior? Does the length of the text matter?
  3. Does the type of writing, fiction vs. non--fiction, prose vs. poetry, etc. change the expected behavior? What pieces of the analysis is this consideration most relevant for?
  4. Are there phrases that appear multiple times in the output for some of the chains? Is it always true that larger values of $n$ lead to more realistic outputs? How are these questions related?
10/20 No Class
No Class.

7 10/23 Lecture Notes

Construct the transition matrix for the random Tower of Hanoi game with two rings and three pegs.

10/25 Lecture Notes

Consider the following time series: $\{-0.0475,-0.0185,-0.3178,-0.2189,0.0164,0.4838,0.3183,-0.3729,0.1841, 0.4489\}$. What are the distributions of patterns of length 2, 3, and 4 that appear? Do these numbers appear to have been drawn from a random distribution?

10/26 X-hour
10/27 Lecture Notes

No homework for Monday.

No quiz this week.

8 10/30 8.1
Construct your ego network (with at least 5 nodes) and compute at least two of the local measures that we discussed in class (code is linked on the resources page). Any interesting observations?
11/1 8.2

No Homework for Friday.

Due: 11/13

Goal: Write a 2-3 page essay analyzing the fission in a social network.

Data and Code: Download link

Details: Zachary's karate club is a standard example of a social network originally analyzed in the 1970's, usually in the context of fission in social groups. The nodes represent members of a university Karate club and the edges represent social relationships between them. The MatLab file containing the adjacency matrix is attached here and you can find Zachary's original research paper at: this link (don't read the paper yet).

Your task has two components. First, you will perform an analysis of the network, using the tools that we have discussed in class, focusing on how the network could split into groups and who the leaders of those groups might be. Secondly, you will analyze your results in light of the contents of Zachary's original research. Your essay should explain your original hypotheses and your reaction to and analysis of the methods and conclusions presented by Zachary. Critically, you should complete the first part before actually reading the paper, so as not to prejudice your initial work.

To help you along the way, here are some questions that may help guide your analysis. Discussion of these need not appear in the essay you turn in but will likely help you perform a thorough analysis on which to base your writing.

  1. Are some nodes more important than others? How can we tell? How do the different centrality measures we discussed in class impact the community structure? We have discussed both local and global measures for centrality, how do these differences impact the analysis?
  2. How does this network compare to the null models that we studied in class? Can you estimate reasonable values for the models to generate examples that look like this network? What does this tell you? Does modularity or assortativity tell you anything interesting?
  3. What is the problem Zacharyis trying to address? What is the anthropological model behind the question? What is his network model and how does it reflect the anthropological model? What assumptions does he make?
  4. What is the network analytic method that Zachary uses? How does it differ from your methods? How does your analysis compare to the author's? What are the positive and negative aspects of the different methodologies?
11/3 8.3

Calculate the assortativity of the dolphin and otter networks.

9 11/6 Lecture Notes

Read the first four pages of this paper: link. Don't hand anything in.

11/8 Chapter 11

Read the beginning of Chapter 11 of your textbook until things stop making sense.

10/26 X-hour
11/10 Lecture Notes

No homework for Monday.

Quiz solutions will go here.

10 11/13 12

You can download the final exam here: link. Your solutions are due on 11/21 at 11am. The exam is open book and open notes (and open MatLab) but the Honor Principle requires that you neither give nor receive aid on this exam. In particular, you may not discuss the problems or solutions with anyone else, although you may ask me questions for clarification.

Daryl DeFord
Last updated November 09, 2017