Research Paper Writing: into the cursing stage….

I am attempting to end a publication drought; I still have one paper at a referee and I am currently writing a more research level paper (it deals with the subject of “wild knots” in 3-space: these are topological homeomorphic images of the circle, but these embeddings are NOT “equivalent” to any differential embedding of the circle in 3-space; it is “no fair” doing calculus in this setting! 🙂 )

I *think* that I have a good idea on how to procede and the first bits have been typeset with no difficulties. But now comes the “cursing” part: I am trying to organize an infinite number of “maps” in a way that allows one to compose these maps in a well defined way.

The frustrating part is that I have to be precise in my language, and at times, precision tends to bury and obscure the idea. People who teach undergraduate mathematics for a living sure understand that! 😉

But the precision is necessary; I remember spending 2 years trying to prove something that was “obviously true”; it turns out that it was false. But I was able to publish the counter-example, and even the reviewer thought that this example was counter-intuitive.

But that was many years ago….so……bring on the cursing!!!!!!

Enigma Machines: some of the elementary math

Note: this type of cipher is really an element of the group , the symmetric group on 26 letters. Never allowing a letter to go to itself reduced the possibilites to products of cycles that covered all of the letters.

Math and Probability in Pinker’s book: The Better Angels of our Nature

I am reading The Better Angels of our Nature by Steven Pinker. Right now I am a little over 200 pages into this 700 page book; it is very interesting. The idea: Pinker is arguing that humans, over time, are becoming less violent. One interesting fact: right now, a random human is less likely to die violently than ever before. Yes, the last century saw astonishing genocides and two world wars. But: when one takes into account how many people there are in the world (2.5 billion in 1950, 6 billion right now) World War II, as horrific as it was, only ranks 9’th on the list of deaths due to deliberate human acts (genocides, wars, etc.) in terms of “percentage of the existing population killed in the event”. (here is Matthew White’s site)

But I have a ways to go in the book…but it is one I am eager to keep reading.

The purpose of this post is to talk about a bit of probability theory that occurs in the early part of the book. I’ll introduce it this way:

Suppose I select a 28 day period. On each day, say starting with Monday of the first week, I roll a fair die one time. I note when a “1” is rolled. Suppose my first “1” occurs Wednesday of the first week. Then answer this: “what is the most likely day that I obtain my NEXT “1”, or all days equally likely?”

Yes, it is true that on any given day, the probability of rolling a “1” is 1/6. But remember my question: “what day is most likely for the NEXT one?” If you have had some probability, the distribution you want to use is the geometric distribution, starting on Thursday of the next week.

So you can see, the mostly likely day for the next “1” is Thursday! Well, why not, say, Friday? Well, if Friday is the next 1, then this means that you got “any number but 1” on Thursday followed by a “1” on Friday, and the probability of that is . The probability of the next one being Saturday is and so on.

The point: if one is studying the distribution of events that have a Poisson distribution (probability ) on a given time period, the overall distribution of such events is likely to show up “clumped” rather than evenly spaced. For an example of this happening in sports, check this out.

Anyway, Pinker applies this principle to the outbreak of wars, mass killings and the like.