2004-12-14: Office hours before the exam. I will be available in my office Wednesday and Thursday 13–16.
2004-12-09:
Where are the results from exercise B and the modelling seminar?
Answer: I got drowned in other tasks, so they're not done.
But now I have some time. So they should arrive Real Soon Now.
2004-11-26: Want to compete in mathematics? Or just learn about problem solving? NTNU is planning to send a team to the 12th International Mathematics Competition for University Students next July. A course in advanced problem solving will be held in the spring. If you're interested in participating in the course (and perhaps being on the NTNU team), send an email to the address given on the announcement page.
Results from problem B and the modelling seminar will be available soon, hopefully on Monday. I will also add a more detailed reading list.
2004-11-15: The modeling seminar is now in session. The problem is briefly described as follows: Many people live along major rivers. No wonder: The soil is fertile, and the landscape is flat and pleasant. However, living in such a place means you are living in a flood plain. The river will flood from time to time. If people are lucky they get water in the basement. If they are less lucky, they lose their homes, or farmers lose their fields and hence theire livelihood. To protect themselves, people build dikes so a bigger than normal river will not flood their land. However, experts say that this makes the flood much worse farther downriver, since the flood water now has only one place to go: Down the river! That is, until it gets so high it overtops the dikes. Therefore, these experts suggest that dikes should not be built, with the possible exception of a few strategically placed ones to protect towns. Your task is not to take a stand in this debate, but to suggest models that can help to assess the effects of dikes (or the lack of such) on a flood, presumably as a tool to help in decision making. (The rest is politics, and that is much too hard for us.)
If you come up with a model that can be solved by hand, that is fine, but it is not required. A hand solvable model will be too simple to fit reality, but we can hope that it will explain what is going on in a qualitative way. You might give some thought to what a full featured computer model might include, but this is secondary. Concentrate first on what you can do with the simplest possible means. (But not too simple, to paraphrase Einstein.)
The groups will present their results on Wednesday next week. This presentation is part of the learning process: I want you to learn from each other. You will not be evaluated by the presentation.
A written report is to be handed in on Thursday next week. Put the student numbers of all the group's participant on the report. Aim the report at people with about the same level of knowledge as yourselves (before you started the project).
The groups. Fewer students showed up than I had expected, so my initial six groups shrank to four. As a result, these groups are probably large enough by now. Any further students should try to form new groups.
Where do we meet this week?
Today, in R52 and R53.
Tuesday (during the regular exercise hour),
in R80 and R81
(D5-133 and D5-137,
on the fifth floor of block D in «realfagbygget»).
Wednesday (during the regular lecture hours),
in R53 and R59
(A2-121 and B2-105,
on the second floor of blocks A and B in «realfagbygget»).
In each case, groups 1 and 2 meet in the first room mentioned,
and groups 3 and 4 in the second room.
2004-11-15: I just remembered that I had promised to post the details on some misprints in the compendium. They are in the next-to-last part, section 5.4 near the end. First, on p. 53 the isotope is said to be shortlived, but that is not quantified. How shortlived? A half-life of about 10 hours seems to fit what comes later.
In formula (254) on the bottom of p. 54 the t1 just after the equality sign should be t0. (The other t1 in that formula is correct.) The formula is also missing a closing parenthesis, but you can figure out where that goes. This error is found again in equation (249) on the middle of p. 55: The first t1 in that equation should be t0.
2004-11-09: I made a note with some of the hairy details concerning the hydraulic jump in the sink. In A4 format (good for printing – pdf, ps) or A5 format (good for screen viewing – pdf, ps). Please note that the note is a bit rough around the edges and may contain misprints and mistakes.
Next week is the time for the modeling seminar. We meet in the usual place, at the usual time on Monday so I can present the problem you will be working on. That will not take very long. Then you will be split into working groups, and we move to more suitable rooms (which I have not yet reserved). The rest of the scheduled time on Monday, Tuesday and Wednesday will be devoted to work on this project. (The groups need to find some more working time, but the rest of the time you will be on your own.)
2004-11-01: I am not putting up a new exercise set this week. Instead, concentrate on exercise set B or, if you're done with it, you may wish to spend some time to review old exercises, etc.
Today I talked about Reynolds' transport theorem. (It was named after the same Osborne Reynolds after whom the Reynolds number is named.) The main use of the Transport Theorem is to allow you to write up a conservation law in its most natural setting, using a material region – i.e., in the Lagrangian viewpoint – and then translate it to a setting sometimes more suited for computation, using regions fixed in space – the Eulerian viewpoint.
I rounded off by showing how the Transport Theorem is used to derive the equations for mass conservation and impulse conservation, with the forces acting on a pipe bend as an example. In the process, I also did mention the material derivative, being the partial derivative with respect to time in the Lagrangian view, and its relationship with the common (Eulerian) partial derivative.
On Wednesday we will deal with the stress tensor and energy conservation, and mention the differential form of the various conservation laws. We'll have a look at the hydraulic jump (see picture).
2004-10-27: I spent more time than I had planned wrapping up our discussion of the scalar conservation law ρt+f(ρ)x=0 (including the traffic example). I thought I received rather strong "vibes" indicating that more explanation was needed. So I just barely got started on the bit about conservation laws in continuum mechanics, did not even have time to write up the Reynolds transport theorem. Oh, well. That must be where we start on Monday, then.
2004-10-26: The second counting exercise set (B) is ready.
2004-10-25: Talked about shock structure today, and started on the road traffic example. On Wednesday, we will wrap up the example (I am not going through everything in the notes), then move on to conservation laws in continuum mechanics.
2004-10-18: Conservation laws this week. Today's lecture was mostly about how to write up a conservation law and what it says. On Wednesday, I will concentrate on the single conservation law in one space dimension, with traffic flow as an example.
2004-10-11: Corrections to the exercise solutions. A sign error has been corrected in the solution to exercise 2 in exercise set 3: (The minus sign on x0=−9/2 had been left out when it was substituted in the ε coefficient.)
Also, the solutions for exercise sets 4 and 5 were for last year's exercises (oops).
This week's lectures are devoted to population models.
Next week, we will get started on conservation laws.
2004-10-01: The "A" exercise is to be turned in on Monday. Look at the exercises page for details and hints.
2004-09-29: No lectures next week. Next week (week 41) is reserved for other activities. Instead, I plan to issue a larger than usual problem set covering all we have done so far.
During the normal lecture hours I will be in my office. You are welcome to drop in and ask questions relating to the course.
In today's lecture I mentioned a few things, such as structural stability and Hopf's bifurcation theorem, that are not in the compendium. I ought to write a short note. I'll let you know when (if) they're done. I showed a graph with a slight variation on the standard Hopf bifurcation. I attach the Maple worksheet I used.
We are now essentially done with the bit on dynamical systems from Logan's book.
2004-09-28: A look at the Lorenz attractor. In the lecture on Monday, I mentioned the Lorenz attractor in passing. This is a strange attractor, exhibiting behaviour that cannot occur in two space dimensions. I wrote a tiny Maple worksheet that produces a graph of the Lorenz attractor which you can then view from different directions to get a better notion of its strangeness.
2004–09–20: Two new exercise sets. The first (number 5) is very small and hopefully routine. It is, as the others up to now, voluntary.
The second (called A) counts 10% for your final grade. You have two weeks on it; I will provide more details later on how I want your answer written up and delivered. Feel free to discuss this with your fellow students, but I want everyone to turn in an answer written in their own words (and formulas).
(This is not quite as I said it would be, but hopefully close enough.)
2004–09–13: Exercise 4 is finally available. I intend to let Exercise 5 be one of the two individual exercises that will count towards your grade.
Today, I finished the enzyme kinetics example and started on stability (and bifurcations). On Wednesday we will look at the slime mold example, and look at bifurcations for a single equation.
2004–09–08: Almost, but not quite, done with the enzyme kinetics example today. The Maple worksheets I used are available: With output removed, for Maple 9 or Maple 8, or (with output intact) converted to PDF.
Next week we'll start on the material from Logan's book on Stability and bifurcation.
I regret to announce that the next exercise set will be delayed. I will be away the rest of the week, for the annual meeting of the Norwegian Mathematics Council, so there is no time left over. I will have it on the web on Monday, however.
2004–09–07: I have corrected the mistakes in the notes I posted yesterday. Follow the links below to get the updated version.
2004–09–06: Today, I went through an example of a singular perturbation problem: The toy problem εy''+y'+y=0 with boundary data y(0)=0, y(1)=1. I even did this to the first order in ε. I did not yet have time to write this up (I made a mistake in the end which ruined the matching), but I will do so soon.
On Thursday, I will look at an example of
a singular perturbation problem: Enzyme kinetics.
Here is a description of the problem:
In A5 format (good for reading from a screen;
ps, pdf)
and in A4 format (good for printing;
ps, pdf).
Note that this note has not been proofread yet,
so it may be full of misprints.
If you wait a little bit, a better version may appear.
(I put the note up anyway because I made a promise.)
2004-09-03: As you can see above, the compendium is now for sale at the department office (4th floor, Sentralbygg 2 / Central Building 2).
Next week I will talk about singular perturbations, after a brief introduction at the end of this week's Wednesday lecture.
2004-09-01: Here is a copy of the Maple worksheet I used in the lecture today. I cleaned it up a little and added a few comments. With output removed, for Maple 9 or Maple 8, or (with output intact) converted to PDF.
2004-08-25: Here is a writeup of the similarity solutions for the heat equation that I went through at the end of today's lecture: As PDF (A5, A4) and as PostScript (A5, A4).
On monday I will talk about a model for the kidney. Here is an extremely brief note. It is in Norwegian, but I will translate it to English as soon as I can find the time: As PDF and PostScript.
2004-08-17: The collision with another class on Monday turns out to be due to a misunderstanding. We will be in F2 in the future, just as planned.
2004-08-16: Slides from today's lecture are available as pdf: with one slide per page, or two slides per A4 page.
Concerning the collision with another class today: I did not have the opportunity to find out who or what class we collided with. If any of you know this information, please let me know so I can contact the teacher of the class and work out a solution. (I have been searching the study handbook to no avail.)
2004-08-13: Several foreign students have asked me to give the class in English. That is what I plan to do. However, since some of the teaching material is in Norwegian, some adjustments need to be made. I need a little time to figure out what and how.
2004-07-20: (I won't bother to translate this to English. It's not that important.) Årets evaluering er litt forskjellig fra tidligere: Modelleringsøvingen i grupper teller noe mindre, og det er kommet til et par individuelle øvinger. Jeg har ennå ikke planlagt detaljene, slik som når i semesteret disse tingene skal skje. Men det blir i hvert fall ikke i løpet av de første tre ukene.
Jeg kommer til å være på ferie fram til 9. august, så eventuelle feil og mangler i denne websiden blir neppe rettet før da. Jeg vil nok lese mail sporadisk i løpet av ferien, så jeg kan prøve å svare på eventuelle spørsmål.