A not so organized guide to The Organized Mind

I made the drive between Rhode Island and Florida two and a half times now. On the very first trip up, I listened to the Organized Mind by Daniel J. Levitin. On the most recent drive, I unwittingly borrowed it again from the library, thinking I’ve never read it before. This should be an indication of how much impact this book has.

To be fair, the first part of the book is quite illuminating, with actual neurological and psychological results on how attention works. With this in hand, he gives practical advice on how to structure one’s work and life, with the biggest theme being “export work out of brain and onto physical space.” The book could have ended after part 1, and it would still be a best seller I bet.

But then, the book divulges into an awkward mishmash of applied math and medical advice. In the section discussing Bayes’ rule, I almost felt that he had a personal vendetta against MDs with many anecdotes of “hurr durr, doctor know no math, dumb.” What was the point of this section? I doubt many people know how to perform a literature check on the efficacy of treatment. Later on, while discussing structure of organization (why?),  he seemed to pen in Shannon’s information theory just to say that the number of levels grows logarithmically (again, why bring it up?).

All in all, read the first part. Disregard rest.

Salt Sugar Fat

I listened to the audiobook version of  Salt Sugar Fat: How the Foot Giants Hooked Us recently, and it’s quite a depressing listen. The author very nicely describes how the processed food industry managed to hook most of the US population on a diet of unhealthy, albeit delicious, foods.

Ultimately, it seems to reside in biology: humans are really bad at living in the modern world. I learned that while there’s a “bliss point” where additional sugar causes actual less enjoyment, no such point exists for fat. That’s a terrifying thought, and one that I have encountered in my own home cooking. The existence of a theoretical maxima for sugar is not a place to anchor one’s hope either; sugar is far less filling and can be incredibly addictive to the point where symptoms of withdrawal can arise.

Another culprit it seems is just greed. The goal to capture more market share results in the manufacturers inventing new ways of capturing the American eye, nose and mouth. The easiest way to do that is the infusion of salt, sugar and fat onto the preservative-laden food, without any regard for the well-being of the consumer. The author constantly makes comparisons with the beleaguered tobacco industry, and it does give the reader glimmers of hope that maybe legal action can help alleviate some of the obesity crisis.

All-in-all, the book was a bit repetitive in some of its material, but still quite interesting.

 

Eigenfunctions and Eigenvalues of the Laplacian of the “Pacman” Domain

We will derive eigenfunctions and eigenvalues on a Pacman domain, which in polar coordinates is $\Omega = \{(r, \theta) : r \in [0, 1], \theta \in [0, 3\pi/2]\}$.
The problem is
\begin{align*}
-\Delta u &= \lambda u \qquad \Omega\\
u &= 0 \qquad \partial \Omega
\end{align*}

In polar coordinates, the Laplacian is
\begin{align}
\Delta = \frac{\partial^2 }{\partial r^2} + \frac{1}{r} \frac{\partial}{\partial r} + \frac{1}{r^2} \frac{\partial^2}{\partial \theta^2}.
\end{align}
Thus, using separation of variables $u(r, \theta) = R(r) \Theta(\theta)$ where $R(1) = 0, \Theta(0) = \Theta(3\pi/2) = 0$, we have
\begin{align*}
\Delta u &= \Theta R” + \frac{1}{r} R’ \Theta + \frac{1}{r^2} R \Theta ” = -\lambda R \Theta.
\end{align*}
Simplifying, we have
\begin{align}\label{eqn:sum0}
\frac{r^2 R” + r R’ + \lambda r^2 R}{R} + \frac{\Theta ”}{\Theta} = 0.
\end{align}
In order for the above to be satisfied, we need each term to be constant, so assume that
\begin{align*}
\frac{\Theta”}{\Theta} = -\lambda_\theta
\end{align*}
where $-\lambda_\theta$ is a constant.
Taking into account the boundary condition, we know that
\begin{align*}
\Theta(\theta) = \sin\left(\frac{2}{3}n \theta \right)
\end{align*}
and $\lambda_\theta = \frac{4}{9}n^2$ for $n \in \mathbb{Z}$.

Now, using (2), we have the corresponding ODE for the $R$ variable
\begin{align*}
r^2 R” + r R’ + (\lambda r^2 – \frac{4}{9}n^2) R = 0.
\end{align*}
Let $\rho = \sqrt\lambda r$, then $R_r = R_\rho \frac{d\rho}{dr} = \sqrt\lambda R_\rho$ and hence $R_{rr} = \lambda R_{\rho\rho}$, hence
\begin{align*}
\rho^2 R” + \rho R’ + (\rho^2 – \frac{4}{9} n^2) R = 0.
\end{align*}
By the change of variables, we know that $R(\rho) = J_{2/3 n}(\rho)$ where $J$ is the Bessel function.

It remains to impose the boundary condition $R = 0$ at $r = 1$, so
\begin{align*}
R(\sqrt\lambda r) = J_{2/3 n}(\sqrt \lambda r) \qquad J_{2/3 n}(\sqrt{\lambda}) = 0.
\end{align*}
meaning that $\lambda = \alpha_{2/3 n, k}^2$ for $k \ge 1$, which are the eigenvalues.

Quickdraw Reviews

Selection bias is a well known fallacy in statistic that is epitomized in the following story:

During World War II, the statistician Abraham Wald took survivorship bias into his calculations when considering how to minimize bomber losses to enemy fire. The Statistical Research Group (SRG) at Columbia University, which Wald was a part of, examined the damage done to aircraft that had returned from missions and recommended adding armor to the areas that showed the least damage, based on his reasoning. This contradicted the US military’s conclusions that the most-hit areas of the plane needed additional armor. Wald noted that the military only considered the aircraft that had survived their missions; any bombers that had been shot down or otherwise lost had logically also been rendered unavailable for assessment. The holes in the returning aircraft, then, represented areas where a bomber could take damage and still return home safely. Thus, Wald proposed that the Navy reinforce areas where the returning aircraft were unscathed, since those were the areas that, if hit, would cause the plane to be lost. His work is considered seminal in the then-nascent discipline of operational research.

While shopping for quickdraws, whose quality is critical to the safety of climbers, there was a product on REI with a good bunch of 5 star reviews with one that stated “I did not die when using it.”

I ended up buying it.

Hopefully, there isn’t a heavy case of selection bias in quickdraw reviews.

The Anime High School

The fact that Cobra Kai is the most popular show on Netflix right now is surprising. I couldn’t help but characterize the show as Americanized anime without the animated aspects (of course) or the gratuitous fan service (thank god). Otherwise, if someone puts a cartoonify filter over the visuals, uses the same dialogue and writes a theme song with actual lyrics, it’s almost indistinguishable from some Japanese anime.

Let’s compare it to the list of tropes from here:

  1. Some Girl’s Unexplainably Huge Boobs Are Going To Be Obsessed Over” – Thank goodness this wasn’t as focus.
  2. “They Go To The Beach Exactly Once” – Check! And even have karate training over water.
  3. Parents Basically Don’t Exist” – Literally no, but figuratively, the adults are generally useless with respect to the kids.
  4. Every Protagonist Starts Off As An ‘Ordinary High School Student'” – Check.
  5. There’s Going To Be A Love Triangle (At Least)” – Check
  6. A Mean Girl Is A Hopeless Romantic” – Season 2, check.
  7. There’s A Club For Everything, And It’s Always Super Important” – Uh, in fact there’s TWO clubs.
  8. There Are Insanely Powerful Student Council Overlords” – Miguel + Hawk/Sam + Robby? Check.
  9. “There’s At Least One Scene Showing Off Traditional School Swimwear” – Okay, this is more a Japanese anime thing…
  10. A Character Who’s Late For School Will Run With Toast In Their Mouth” – WTF. Maybe the corresponding thing in America is being dropped off by the parents?
  11. The Teachers Are So, So Sexy” – Thank god no.
  12. One Person That’s Over-Determined To Have A Rival, Who Just Happens To Be The Main Character” – LITERALLY THE WHOLE PLOT IS KIDS/ADULTS LOOKING FOR AN ENEMY JUST BECAUSE.
  13. A Girl Gets Scarily Obsessed With Someone” – Season 2 finale anyone?
  14. They’ll Have Insane Names For Sports Tricks” – Nope.

So out of fourteen, there’s a total of nine. Of the five tropes that aren’t in Cobra Kai, three of them are the stupid sexual fan service nudity…

Probably why it’s so addictive.

Musings on Avatar

With the Legend of Korra on Netflix, I’ve re-watched the entire Avatar animated series twice now during quarantine. For a show aimed at kids, the two series are known to be surprisingly deep both in character development (the strength of the original series) and conflicts (where Korra really shines). Here are some incoherent thoughts that have been swirling inside my head:

  1. With the use of lightning benders as power generators in Korra, one wonders what sort of physical laws govern the Avatar universe. It is mentioned that radio and EMF exist in the universe alongside internal combustion energy. The more interesting question to me is the issue of conservation of energy, and if so, where does the extra energy from lightning and lava benders come from? One possible clue to this is answered in Korra with the Kuvira plot line.

    It turns out that spirit vines contains vast amount of energy which can be harnessed in the physical world, akin to the strong nuclear force in our world. Furthermore, the powers of a bender seems to be distinctly tied to the spirit world: the first benders were given the power by Lion Turtles through energy bending and the avatar’s powers are amplified by the spirit Raava. So if a sort of conservation law were to exist for bending, it must be that the benders are channeling additional energy from the spirit realm. Could it be that this is why there was animosity amongst the spirits and humans initially? And could the energy somehow be filtering back towards the spirit realm through the portals?

  2. Korra touches on a lot of topics that occurred in the real world during the 20th century. One wonders how the avatar and friends would handle a pandemic. Whole episodes can probably be written about this.

    Ideas include: it’s all airbender conspiracy that’s killing people, Future industries is secretly sabotaging the republic by poisoning the radio, the spirits are actively killing people because *insert any arbitrary reason*

  3. In the Avatar universe, there’s no such thing as equality among races. Water benders definitively cannot fire bend and vice versa. Similarly, it’s also implied that genetics play a huge role in the power level of a bender: the stronger your parents were, the stronger you likely will be. I suspect this is part of the reason why the royals of the water tribe and fire nation was so strong in their bending (this begs the fact why the earth kingdom’s monarchy was so inept at the time of the show).

    Therefore, the great man theory should be the prevailing theory in that universe’s history. The level of change one single powerful bender can have over local events is huge. Need to conquer an opposing village? Just send your strong bender and wipe them out. And let’s not even talk about the overwhelming power the avatar wields. It’s a world where being born without any bending skills means a life closed off from many jobs and employment. Honestly, this is one of the reasons why being born in the avatar universe seems miserable at times.

Reproducibility of Numerical Data

Using git to version control my code and LaTeX files for my academic career is probably one of the best habits I have.  I do sometimes slack off and have conflicts between the repository on my work computer/laptop/home computer which is agonizing to resolve, but those instances are rare. In general, I try to commit/push once a day with a short comment on what was accomplished, and this allows me to backtrack to a great extent if needed.

Unfortunately, there seems to be two issues which I’m seeing right now while revising a paper. The first is that I should also state the version of the auxiliary software and packages which my own code depends on. I found out this the hard way when I noticed that different versions of gmsh resulted in different meshes, even on simple domains such as the cube. I believe this can be simply resolved with a setup.py or a requirements text file.

The other is that I need to record the exact code and parameter configuration when presenting data. What this entails is to commit code every time data is added to the write up. Then, I should also add the commit ID to the LaTeX file.