Our study of Conway's 23/3 rule in HW#8 may have some bearing on the current debate over evolution in Kansas as described in this New York Times article. What do you think?

Solutions.

Exam #3 is on Thursday, April 28, and covers everything we have discussed since we started talking about chemistry and the principle of mass action. Exam is closed-book: no notes this time. (Please commit the heat/diffusion equation, and the wave equation to memory if you haven't already!) Bring a non-programmable, non-graphing, scientific calculator.

See Diary for an overview of topics. See your class notes, Coursepack and updated Text 3a for detailed information. I will have office hours Monday and Wednesday from 1-2pm, and we will spend class time Tuesday reviewing.

Diary

Gradesheet

Semester review

Some new software called STEM for modelling the spread of epidemics in the US has just been released by IBM researchers. You will find familiar some terms in the press release. (In case that disappears, a cached version is here.)

Diary

Exam #3 is on Thursday, April 28, and covers everything we have discussed since we started talking about chemistry and the principle of mass action. See Diary for an overview of topics. I will have office hours Monday and Wednesday from 1-2pm, and we will spend class time Tuesday reviewing.

Homework #8, due Thursday, April 21. Solutions.

Homework #7, due Thursday, April 14. Solutions.

Homework #6, due Tuesday, April 5. Solutions.

Exam #2 is this Thursday, March 24. You may bring one 5"x8" hand-written (non-xeroxed) note-card to the Exam. You may also bring (and may need) a non-programmable, non-graphing scientific calculator. The exam will cover eveything from "deterministic vs. random models" on Feb 8 through to the "principle of mass action" on March 10: see the Diary for an overview. You should not be surprised to see questions on, for example, formulation (and solution?) of ODEs for probability evolution in some generalization of the radioactive decay and birth process models we have studied; construction of a sequence of random numbers with a specified distribution from a uniformly distributed sequence; construction of a solution of the heat equation using the fundamental plane source solution as a building block (method of images?); finding the temperature phase lag or lead of a sinusoidally heated slab (here is the completion of the finite-slab problem (corrected 3/25/05) we started in class. Various other questions on the specified material may also appear.

Exam #2 Solutions

Homework #5, due Tuesday, March 22. Solutions.

Homework #4, due Thursday, March 3. Solutions.
In the news: Experts say a pandemic is inevitable and will probably emerge in Asia if bird flu mutates with human flu, creating a highly infectious new virus.

Homework #3, due Tuesday, Feb. 22. Solutions
Exam #1 is closed-book, with no notes allowed. You'll need a non-programmable, non-graphing scientific calculator. Problems will largely be similar to some of those seen in HW#1 and HW#2. Topics very likely to appear are fixed and periodic points of 1D maps and their stability, and asymptotics of linear age-structured models. The stochastic processes we've been discussing this past week are NOT on Exam #1. Exam #1 Solutions.
Homework #2, due Thursday, Feb. 10. Solutions: scanner version, camera version.
Homework #1, due Tuesday, Feb. 1. Solutions.

Gradesheet Note on HW7 grades: I reviewed the grades on HW7 4/22/05 and found many errors by the grader in adding scores up. You may therefore find that your score has been adjusted down, but note the max possible is down from 29 to 27. FYI, in Problem 1d, 3 points were for your written answer, and 2 extra-credit points were given if you sent photos.

WED, APRIL 27, in 206 Furnas Hall at 3:30pm
Stanislav Y. Shvartsman
Princeton University
PATTERN FORMATION IN DEVELOPING EPITHELIAL LAYERS
More info.

Here are the experimental results I have received from students so far.
Examples for class 4/21/05

gray_scott.mws

Here are stripe pix you sent me:

Here is the latest update of Text 3a. Please note you have to remove part of the file name when you get "Not found" message. (This is to prevent this draft document from being slurped by Google, etc.)

Brusselator stability boundary, real roots case: mws
Brusselator stability boundary, complex roots case: mws

Here is a Java applet I wrote that uses Euler's method to solve the reaction-diffusion equations for the Brusselator. You can play with the parameters to observe a variety of temporal and spatial patterns.

Compare the steady patterns in the Brusselator with other things you've seen, like this: .

Continuous functions can be built out of sinusoids:

Animation of Fourier synthesis.

The next series of courses (Fall2005) in the Peter Canisius Program "New Mathematical Topographies" will concentrate on mathematical biology and anthropology. These are three one- week one-credit courses that students, staff & faculty may take for credit, audit, or simply attend in an informal way. Each course is taught by a *high profile* visiting scholar.

Dr. Eduardo Mercado III, Mathematical Models of Singing Whales (MAT 317)
Dr. Per Enflo, Mathematical Models in Biology & Anthropology (MAT 318)
Dr. Robert E. Kass, Statistical Analysis of Neuronal Data (MAT 319)
More info here

Solar heating of the Earth: mws.
Animation of exp(i &omega t): GIF, mws. (Annoyingly, after all these years Maple still cannot fill a polygon correctly. And the colors are messed up in the GIF export.)

Fundamental plane source solution and solution for sudden change of surface temperature: animation, mws.

Text 1d pdf

In case you're interested, here is the answer to Jake's question about stability of a fixed point when the slope = -1. We can determine what happens by looking at f². When f(x) = x and f'(x) = -1, then a little calculation shows that
f²(x) = x,
f²'(x) = 1,
f²''(x) = 0, and
f²'''(x) = -3 (f''(x))² - 2 f'''(x).
Therefore the fixed point is repelling if -3 (f''(x))² - 2 f'''(x) > 0, and attracting if -3 (f''(x))² - 2 f'''(x) < 0. That is to say, repelling if -3 (f''(x))² > 2 f'''(x), and attracting if the opposite is true. Note that if f'''(x) > 0, then it's attracting for sure. Example cobwebs here.

Notes paralleling Thursday's lecture are here: Probability pdf. (Note that I have altered the name so that this rough draft is not picked up and cached by search engines.)

Simulating radioactive decay: mws.

It's probably best if you use the new LinearAlgebra package instead of the deprecated linalg package. Here's how to use LinearAlgebra in HW2. If you have any questions about Maple, be sure to let me know.

In case questions are Maple-related, I will hold my office hour on Monday Feb 7 in Hochstetter 139.

Iterating a linear age-structured model. Model#1 (4-stage): mws, pdf, html (contains largish animated GIFs). Model#2 (81-stage): mws, pdf No HTML, sorry: Maple 9.5 hangs when trying to export this file).
Animation only

Another Help Session for Maple Beginners and other interested students: this Monday, Jan 31, 12-12:50pm, Hochstetter 139.

Help Session for Maple Beginners and other interested students: Friday, Jan 28, 9-9:50am, Hochstetter 139. We will work through this set of exercises. It would be great if you could work through the at least the first 4 sections of the guide below before the session. Remember: Maple is available for Linux, Mac, and Windows for less than $2 at UBMicro in the UB Commons.

The HTML files can be opened with any browser. You need Maple to open the mws files.
Maple as calculator: HTML, mws.
Maple as analytical tool: HTML, mws.
Maple as programming language: HTML, mws.
Maple for pictures: HTML, mws.
Maple for 3D pictures: HTML, mws.

Unsurprisingly, there is no time when everyone is available. Here are the aggregated responses: your unavailable times. Based on your responses, I have chosen Mondays 1-1:50pm, and additionally the Tuesdays or Thursdays on which HW is due from 9-9:30am.

Mayfly: a species with non-overlapping generations.
Notes from Day 1.
Struggle for existence.
Cobweb diagrams with Maple (Day 2 solutions).
Logistic map.
Conjugacy of logistic and tent maps.

Assignment: exploring the dynamics of the logistic map (PDF).
An example of the descriptions you turned in:

It would be helpful if you can check that you can log into the nsm system BEFORE class Thursday (1/20). From any system try to "ssh" or "telnet" into unix.nsm.buffalo.edu, or go to the always-open labs Bell 101 or Hochstetter 141 and see if you can login. I have requested accounts for all students in our class, but if you seem not to have an account on the NSM system, or you do not know your password, please contact accounts@eng.buffalo.edu.

Some features of this site may not be properly rendered by MS Internet Explorer. A W3C standards-compliant browser such as Firefox (free download) is recommended.

Course handout: PDF, HTML
Tentative schedule of topics

We will use a collection of texts by several authors, compiled in a Coursepack which is available from Great Lakes Graphics and Printing in the UB Commons. A confirmation will be posted here that it is ready. You will need a 3-ring binder (1" should be thick enough) to hold it.