Category Archives: Story

Daily Math in Zagreb

So I’m in Zagreb now, and naturally this means that I’ve not updated this blog in a while. But this is not to say that I haven’t been doing math! In fact, I’ve been doing lots, even little things to impress the girl. ‘Math to i-impress the g-girl?’ you might stutter, a little insalubriously. Yes! Math to impress the girl!

She is working on finishing her last undergrad thesis right now, which is what brings us to Croatia (she works, I play – the basis for a strong relationship, I think… but I’m on my way to becoming a mathematician, which isn’t really so different to play). After a few ‘average’ days of thesis writing, she has one above and beyond successful day. This is good, because she is very happy on successful days and gets dissatisfied if she has a bad writing day. So what does a knowledgeable and thoughtful mathematician do? It’s time for a mathematical interlude –

Gambling and Regression to the Mean

There is a very well-known fallacy known as the Gambler’s Fallacy, which is best explained through examples. This is the part of our intuition that sees a Roulette table spin red 10 times in a row and thinks, ‘I bet it will spin black now, to ‘catch up.’ ‘ Or someone tosses heads 10 times in a row, and we might start to bet that it’s more likely than before to toss tails now. Of course, this is fallacious thinking – neither roulette nor coins has any memory. They don’t ‘remember’ that they’re on some sort of streak, and they have the same odds from one toss to another (which we assume to be even – conceivably the coin is double-sided, or the Roulette wheel is flat and needs air, or something).

The facts that flipping a coin always has about even odds and that the odds of Roulette being equally against the gambler are what allow casinos to expect to make money. It also distinguishes them from games with ‘memory,’ such as blackjack (I happen to think that Bringing Down the House is a fun read). But that’s another story.

But the related concept of ‘Regression to the Mean’ holds more truth – this says that the means of various sets of outcomes should eventually approximate the expected mean (perhaps called the ‘actual mean’ – flipping a coin should have about half heads and half tails, for instance). So if someone flips a coin 20 times and gets heads all 20 times, we would expect them to get fewer than 20 heads in the next 20 throws, Note, I didn’t say that tails are more likely than heads!

Back to the Girl

So how does this relate? I anticipated that the next day of writing would not be as good as the previous, and that she might accordingly be a bit disappointed with herself for it. And, the next day – she was! But alas, I came prepared with sour cherry juice (if you’ve never had it, you’re missing out), and we picked up some strawberries. Every day is better if it includes sour cherry juice and strawberries.

Posted in Humor, Mathematics, Story | Tagged , , , , , | Leave a comment

Integration by Parts

I suddenly have college degrees to my name. In some sense, I think that I should feel different – but all I’ve really noticed is that I’ve much less to do. Fewer deadlines, anyway. So now I can blog again! Unfortunately, I won’t quite be able to blog as much as I might like, as I will be traveling quite a bit this summer. In a few days I’ll hit Croatia.

Georgia Tech is magnificent at helping its students through their first few tough classes. Although the average size of each of the four calculus classes is around 150 students, they are broken up into 30 person recitations with a TA (usually a good thing, but no promises). Some classes have optional ‘Peer Led Undergraduate Study’ programs, where TA-level students host additional hours to help students master exercises over the class material. There is free tutoring available in many of the freshmen dorms every on most, if not all, nights of the week. If that doesn’t work, there is also free tutoring available from the Office of Minority Education or the Department of Success Programs – the host of the so-called 1-1 Tutoring program (I was a tutor there for two years). One can schedule 1-1 appointments between 8 am and something like 9 pm, and you can choose your tutor. For the math classes, each professor and TA holds office hours, and there is a general TA lounge where most questions can be answered, regardless of whether one’s TA is there. Finally, there is also the dedicated ‘Math Lab,’ a place where 3-4 highly educated math students (usually math grad students, though there are a couple of math seniors) are available each hour between 10 am and 4 pm (something like that – I had Thursday from 1-2 pm, for example). It’s a good theory.

During Dead Week, the week before finals, I had a group of Calc I students during my Math Lab hour. They were asking about integration by parts – when in the world is it useful? At first, I had a hard time saying something that they accepted as valuable – it’s an engineering school, and the things I find interesting do not appeal to the general engineering population of Tech. I thought back during my years at Tech (as this was my last week as a student there, it put me in a very nostalgic mood), and I realized that I associate IBP most with my quantum mechanics classes with Dr. Kennedy. In general, the way to solve those questions was to find some sort of basis of eigenvectors, normalize everything, take more inner products than you want, integrate by parts until it becomes meaningful, and then exploit as much symmetry as possible. Needless to say, that didn’t satisfy their question.

There are the very obvious answers. One derives Taylor’s formula and error with integration by parts:

$latex \begin{array}{rl}
f(x) &= f(0) + \int_0^x f'(x-t) \,dt\\
&= f(0) + xf'(0) + \displaystyle \int_0^x tf”(x-t)\,dt\\
&= f(0) + xf'(0) + \frac{x^2}2f”(0) + \displaystyle \int_0^x \frac{t^2}2 f”'(x-t)\,dt
$ … and so on.

But in all honesty, Taylor’s theorem is rarely used to estimate values of a function by hand, and arguing that it is useful to know at least the bare bones of the theory behind one’s field is an uphill battle. This would prevent me from mentioning the derivation of the Euler-Maclaurin formula as well.

I appealed to aesthetics: Taylor’s Theorem says that $latex \displaystyle \sum_{n\ge0} x^n/n! = e^x$, but repeated integration by parts yields that $latex \displaystyle \int_0^\infty x^n e^{-x} dx=n!$. That’s sort of cool – and not as obvious as it might appear at first. Although I didn’t mention it then, we also have the pretty result that n integration by parts applied to $latex \displaystyle \int_0^1 \dfrac{ (-x\log x)^n}{n!} dx = (n+1)^{-(n+1)}$. Summing over n, and remembering the Taylor expansion for $latex e^x$, one gets that $latex \displaystyle \int_0^1 x^{-x} dx = \displaystyle \sum_{n=1}^\infty n^{-n}$.

Finally, I decided to appeal to that part of the student that wants only to do well on tests. Then for a differentiable function $latex f$ and its inverse $latex f^{-1}$, we have that:
$latex \displaystyle \int f(x)dx = xf(x) – \displaystyle \int xf'(x)dx = $
$latex = xf(x) – \displaystyle \int f^{-1}(f(x))f'(x)dx = xf(x) – \displaystyle \int f^{-1}(u)du$.
In other words, knowing the integral of $latex f$ gives the integral of $latex f^{-1}$ very cheaply, and this is why we use integration by parts to integrate things like $latex \ln x$, $latex \arctan x$, etc. Similarly, one gets the reduction formulas necessary to integrate $latex \sin^n (x)$ or $latex \cos^n (x)$. If one believes that being able to integrate things is useful, then these are useful.There is of course the other class of functions such as $latex \cos(x)\sin(x)$ or $latex e^x \sin(x)$, where one integrates by parts twice and solves for the integral. I still think that’s really cool – sort of like getting something for nothing.

And at the end of the day, they were satisfied. But this might be the crux of the problem that explains why so many Tech students, despite having so many resources for success, still fail – they have to trudge through a whole lost of ‘useless theory’ just to get to the ‘good stuff.’

Posted in Expository, Georgia Tech, Mathematics, Story | Tagged , , , , | Leave a comment

An even later pi day post

In my post dedicated to pi day, I happened to refer to a musical interpretation of pi. This video (while still viewable from the link I gave) has been forced off of YouTube due to a copyright claim. The video includes an interpretation by Michael Blake, a funny and avid YouTube artist. The copyright claim comes from Lars Erickson – he apparently says that he created a musical creation of pi first (and… I guess therefore no others are allowed…). In other words, it seems very peculiar.

I like Vi Hart’s treatment of the copyright claim. For completeness, here is Blake’s response.

Posted in Humor, Story | Tagged , , | Leave a comment

A Bag’s Journey in Search of its Owner

This is the story of a bag,
who lost its owner and trav’led the whole world!
And though it left with lots o’ tags attached,
She absolutely lost it, when she flied.

How many days would it be?
She arrived with hope, but found only tears.
The bag just disappeared,
so she flew in without any gear.
But she gets a call the next morning,
“Where are you, your bag is right here!”
Thousands of miles afar.
When she looks in the mirror so how does she choose?
The same clothes worn day after day.
When travelling homeward bound,
her bag seems never to be found.

This is the story of a bag,
who lost its owner and trav’led the whole world!
And though it left with lots o’ tags attached,
She absolutely lost it, when she flied.

[loosely to “Story of a Girl”]

This is one of those strange stories – girl gets ready for flight from Atlanta to New York to Prague, girl ends up going Atlanta to Norfolk to New York to Prague, but bag ends up going Atlanta to New York to Atlanta to New York to Prague to New York to Atlanta to a warehouse to Atlanta to New York to Prague to Krakow… you know, the typical story. To be fair, the flight change through Norfolk as opposed to a direct to New York was last minute, and it makes sense for the bag to have been detained in New York. Perhaps it would make it on a later flight to Prague – such is life.

But nothing so simple occurred. The bag makes it to Prague, and when the girl notes that the bag should be sent to her home, one might expect the story to end. Instead, the bag ends up back in New York, then back in Atlanta. Of course, girl doesn’t know this – it’s all a big mystery (as she borrows friends’ clothing, of course). Fortunately, a Delta worker named Carl (I think) finds this bag and its tag in this warehouse, looks it up and calls girl. Girl asks for it to be shipped to her – no problem, he says. Carl is very good at his job, I think, and I commend him. Unfortunately, the bag gets to Prague again and somehow whatever instructions were once somehow connected to the bag are lost. So now someone at Prague calls up girl – what do you want to do with this bag? So the bag goes to Krakow, but that’s okay. That’s where the girl found the bag.

A very logical route, one might say.

Posted in Humor, Story | Tagged , | 1 Comment

About 3 weeks behind:

On 8 March 2011, Dr. Thomas Weiler and graduate fellow Chiu Man Ho of Vanderbilt put a paper on another possibility of achieving a sort of time travel. This apparently got a good deal of press at the time, as both CBS and UPI actually picked up the story and ran with it.

Why do I mention this? It is most certainly not because I have a dream or hope of time travel – quite the opposite really. In the past, I have talked of how surprised I was at the lack of hype coming out of the LHC. The last terrible bit I heard was a sort of rogue media assault on the possibility of the LHC creating a black hole and thereby destroying everything! But that was years ago and not stirred up by the high energy physics community.

As an aside, it did provide the very comical, which includes a very simple answer and funny source code. By the way, no – as far as we can tell, it hasn’t yet destroyed the world.

Let’s get clear – I don’t think that hype is bad. Dr. Weiler himself noted the speculative interest in this idea and that it’s perhaps not the most likely theory. And it doesn’t contradict M-Theory, apparently. I know nothing of this, so I can’t comment. But I can say that such fanciful papers are wonderful. This sort of free form play is liberating, and exactly the same sort of thing that drew me into science. What can we say about the world around us that goes along with what we know? Whether it’s correct or not is something that can be explored, but it’s just an idea.

For those who don’t want to read the article, Dr. Weiler and Chiu Man Ho allude to the possibility of transferring Higgs singlets (a relative to the as-yet-only-hypothesized Higgs Boson) to a previous time. So no, unfortunately we cannot yet fix the problems of our past.

Nonetheless, there should be more hype about the LHC. The test schedule on the collider is becoming more intense all the time. Very exciting.

Posted in Expository, Story | Tagged , , , | Leave a comment

A late pi day post

As this blog started after March 14th, it hasn’t paid the proper amount of attention to $latex \pi $. I only bring this up because I have just been introduced to Christopher Poole’s intense dedication to $latex \pi $. It turns out that Christopher has set up a $latex \pi $-phone, i.e. a phone number that you can call if you want to hear $latex pi $. It will literally read out the digits of $latex \pi $ to you. I’ve only heard the first 20 or so digits, but perhaps the more adventurous reader will find out more. The number is 253 243-2504. Call it if you are ever in need of some $latex \pi $.

Of course, I can’t leave off on just that – I should at least mention two other great $latex \pi $-day attractions (late as they are). Firstly, any unfamiliar with the $latex \tau $ movement should read up on it or check out Vi Hart’s pleasant video. I also think it’s far more natural to relate the radius to the circumference rather than the diameter to the circumference (but it would mean that area becomes not as pleasant as $latex \pi r^2 $).

Finally, there is a great musical interpretation and celebration of $latex \pi $. What if you did a round (or fugue) based on interpreting each digit of $latex \pi$ as a different musical note? Well, now you can find out!

Until $latex \tau $ day!

Posted in Mathematics, Story | Tagged , | 1 Comment