Fleas and Galton-Watson

Consider an ennemy Galton-Watson process where each node has 20 offsprings on average. If you kill 90% of the nodes, is that a success? No really: the remaining nodes are still a rather impressive number. How to destroy them, then? The solution is to repeat the destructive process a second time. If the second attack kills 90% of the remaining nodes, then you will be left with just a few sparse isolated surviving individuals who can then be killed one by one as they come out of their hiding places.

Apply this to flea infestation: I am lauching my second chemical weapons attack today, with stronger chemicals that leave a residue on floor and furniture, so that any egg that might still hatch in the near future would then be killed as soon as it is born.

Flea war

I have launched a massive chemical weapon attack in my house, ongoing as I write.

Fleas are a good example of an exponential process. One day, you think you might have gotten a bite but are not sure. The next day, you spot a flea. The next day, 5, the next day, your son complains that he can't work on the computer because fleas keep jumping on his ankles. Time to declare war! And that's why my house is currently being filled with toxic gas.

Where do those fleas come from? From a litter of kittens who were in my house for a few weeks. I knew they had fleas, but the fleas politely stayed on them and did not bother anyone else. I half-heartedly tried to bathe them and comb them with a flea comb, but my efforts were ineffective, and because of the youth of the kittens, there was no other possibility. After the kittens were gone, the flea eggs were left behind, and must have hatched in the last few days.

Infinity

Toothpaste I always thought of as an image of infinity: no matter how long you've been using a toothpaste tube, there is always a little bit more that can be squeezed out if you try hard enough. There is definitely a first time, but then you can always use it one more time. You never get to the end.

But last week I reached the end of my tube. It made a sucking noise, a hole appeared in the toothpaste, and, squinting through it, I could see the empty inside of the tube. First time in my entire life. Somehow, it makes me think of death.

Sour patch candies

One sour patch candy loose at the bottom of a backpack +
tablet +
chord +
a few days =
a big sticky mess.

Grading efficiently

Here is how I grade Theory homeworks.
1. Grade everybody's answer to problem 1, then everybody's answer to problem 2, etc. This goes much faster than grading all problems of the first homework, then all problems of the second homework, etc.
2. Read a couple of solutions by students who I know will probably get it mostly right.
3. After that, sketch a model solution; figure out the key points/keywords.
4. When reading a proof, scan it briefly to find the keywords. Ideally, someone using the right keywords has to have done the proof right, and someone missing a keyword has to be missing something in their argument.
5. Grade at a high level. Too much granularity means wasting time on unimportant details; some experiments have also shown, I have heard, that grades of the same exams by independent graders yield (paradoxically) more consistent results if the grading is more global. This means that I have to have a clear sense of "what really matters".
6. Occasionally give up. I typical spend the majority of my time on a tiny fraction of the homeworks (the ones that come up with a completely different approach, or from students who are unusually wordy and obscure, or a combination). Sometimes, if after some amount of time I still cannot figure out what their solution is about, I just add a note in the margin: "No credit -- I can't understand this. Come see me at office hours and if you can explain it to me you may get some credit."
7. Design homeworks so that they are easy to grade by the above method. In particular each question should have a reason for being there: it should test something specific, and then when I grade I am just looking for that key point. Example: ask for a proof by induction, in an application where there is a risk that the student may list the base cases wrong. When grading, I can then scan the whole proof and just zoom on the base cases. The rest is "noise" and I will essentially ignore it, although the student does not know it when he writes his solution. Some students then get annoyed because they will receive almost no credit even though, in their view, they got most of the question right; but they didn't figure out the one part I cared about...