Switching the order of summation

Posted by Jason Polak on 21. March 2017 · Write a comment · Categories: elementary, exposition

Characteristic functions have magical properties. For example, consider a double summation:
$$\sum_{k=1}^M\sum_{r=1}^k a_{r,k}.$$
How do you switch the order of summation here? A geometric way to think of this is arrange the terms out and “see” that this sum must be equal to
$$\sum_{r=1}^M\sum_{k=r}^M a_{r,k}.$$
I find this unsatisfactory because the whole point of good notation is that you shouldn’t have to think about what it actually means. I do think it’s very important to understand what the notation means, but in doing formal manipulations it’s nice not to have to do that all the time.

A superior proof that these two sums are equal would be to write
$$\sum_{k=1}^M\sum_{r=1}^k a_{r,k} = \sum_{k=1}^M\sum_{r=1}^M a_{r,k}\delta(r\leq k)$$
where $\delta(r\leq k)$ is the function of the variables $r$ and $k$ that equals one exactly when $r\leq k$ and zero otherwise. Then we can happily switch the order of summation to get
$$\sum_{r=1}^M\sum_{k=1}^M a_{r,k}\delta(r\leq k).$$
Now, it’s trivial to get rid of the $\delta(r\leq k)$ function by writing
$$\sum_{r=1}^M\sum_{k=r}^M a_{r,k}.$$

Book Review: Jayawardhana’s “Neutrino Hunters”

Posted by Jason Polak on 16. January 2017 · Write a comment · Categories: exposition · Tags:

A neutrino is a ultra low mass chargless subatomic particle that is produced in a variety of nuclear reactions such as beta decay. Neutrinos are incredibly abundant, but because of their size and lack of charge, they pass through almost anything and are extremely difficult to detect. Ray Jayawardhana’s book Neutrino Hunters is a fascinating glimpse into the great strides made by physicists to actually detect and understand these mysterious particles.

Neutrino Hunters progresses historically from Wolfgang Pauli’s hypothesising the existence of neutrino to its experimental confirmation and analysis as a central player in the workings of the universe. Several colourful and intriguing portraits of physicists appear along with the incredible experiments that were devised to understand the neutrino. Particularly fascinating were the early neutrino detectors, filled with hundreds of liters of dry-cleaning fluid that had to be placed deep underground to avoid interference. As time goes on, the detectors become more complex and even more ingenious, though I’ll leave out specifics so as not to spoil the book. Suffice it to say, some of the feats accomplished with complicated apparatus are truly astounding.

After the story of the then-state of the art is told, the author explains some future experiments and unresolved speculations, such as the possibility of a fourth neutrino flavour and neutrino communication.

The author manages to keep an excellent balance between precise scientific explanation for the nonspecialist and lively historical recounting. As someone who usually is terribly bored with history, I was never bored reading this book. This is not surprising, as the author is a physicist himself and does not fall into the habit (usually possessed by journalists without scientific training) of including vast amounts of irrelevant detail.

Overall, Jayawardhana is precise without ever being overly technical, and Neutrino Hunters should be easily readable by anyone with a basic knowledge of the atom and a yearning to glimpse into the subatomic world and the origins of the universe. Highly recommended.

Six Tips for Math Bloggers

Posted by Jason Polak on 07. January 2017 · Write a comment · Categories: exposition · Tags:

Math blogging is a fun part of being a mathematician. For me, it’s an aid to reading literature and an outlet for my writing prediliction. Blogging is cool because you write whatever you want without having to worry about the sometimes arbitrary and muddled standards of publication. But how do you do it? In this post I’ll give six tips on math blogging that should help.

Blogging is such a carefree medium that there’s no reason to worry about failed or unfinished posts. I have a folder of such posts just in case I want to resurrect any of them, and it contains around 80 posts–or just under half of the number of all the posts published. These range from very preliminary to finished and polished, totalling about 48000 words, or about twice the number of words in Hampton Fancher’s script for the movie Blade Runner. And you know what? It was fun to write those too, but in the end I decided they were not the right material for Aleph Zero Categorical.

Surprisingly, it’s actually fine to write about finitely generated abelian groups even though you’re working on interuniversal Teichmuller theory. It’s also fine to have a sophisticated blog. Mathematics actually needs much more exposition, so there’s really no need to restrict yourself to certain topics because you think other mathematicians will look down on it.

Pick a topic and stick with it. For my blog, it’s math and related fields, like computer science and applications, though mostly I just write about algebra. There have been occasions where I’ve been tempted to post reviews of books I’ve read in other fields like biology but I’ve resisted because that was outside the scope I set out, and I doubt it would make sense to my audience. Writing outside the scope is a slipperly slope: first it’ll be one or two posts on chemistry, and then pretty soon you’ll be writing on bizarre topics like politics and South American mushrooms.

4. Update regularly

I require myself to produce one post per month. Only once or twice did that fail in the past five or so years, and on average I’m way above that. Not only will a regular update requirement keep you blogging, but it will keep your readers around. Most math blogs miss months, so I figure I’m safe.

5. Heed the format

A blog post is not supposed to be long, and people don’t visit blogs to read proofs of the four colour theorem. I’ve definitely written posts that were too long, and in the end those were not so popular. Keep posts to a main idea, keep it under a thousand words, and your blog will be far more readable for it.

6. In the end, it doesn’t have to be math

Weird advice for a post on math blogging right? But in the end if you don’t enjoy math blogging, you might still enjoy food blogging or posting pictures of rocks.