Beginning to migrate Wednesday, Oct 23 2013 

I haven’t updated this in a bit, but I have a good reason: I am in the midst of migrating away from wordpress.com to davidlowryduda.com.

This is the first time I’ve dealt with backend-ish things, so it took me a bit to get used to the lay of the land. Although I have the domain davidlowryduda (go figure), I am currently using a free webhost (I’m going to assume that I’m not going to suddenly out-traffic it or anything). So it’s a bit slow compared to wordpress.com, but there will be no more ads! (yay!) And I have complete control over the layout, and I can have a better site.

Further, I’m experimenting with django, and I haven’t yet decided which one I prefer or how I might want to integrate them together.

This is all to say that davidlowryduda.com is in a state of flux, but I’ll be maintaining both this and that for a while – until I get fully set up.

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Happy Birthday to The Science Guy Tuesday, Sep 10 2013 

On 10 July 1917, Donald Herbert Kemske (later known as Donald Jeffry Herbert) was born in Waconia, Minnesota. Back when university educations were a bit more about education and a bit less about establishing vocation, Donald studied general science and English at La Crosse State Normal College (which is now the University of Wisconsin-La Crosse). But Donald liked drama, and he became an actor. When World War II broke out, Donald joined the US Air Force, flying over 50 missions as a bomber pilot.

After the war, Donald began to act in children’s programs at a radio station in Chicago. Perhaps it was because of his love of children’s education, perhaps it was the sudden visibility of the power of science, as evidenced by the nuclear bomb, or perhaps something else – but Donald had an idea for a tv show based around general science experiments. And so Watch Mr. Wizard was born on 3 March 1951 on NBC. (When I think about it, I’m surprised at how early this was in the life of television programming). Each week, a young boy or a girl would join Mr. Wizard (played by Donald) on a live tv show, where they would be shown interesting and easily-reproducible science experiments.

Watch Mr. Wizard was the first such tv program, and one might argue that its effects are still felt today. A total of 547 episodes of Watch Mr. Wizard aired. By 1956, over 5000 local Mr. Wizard science clubs had been started around the country; by 1965, when the show was cancelled by NBC, there were more than 50000. In fact, my parents have told me of Mr. Wizard and his fascinating programs. Such was the love and reach of Mr. Wizard that on the first Late Night Show with David Letterman, the guests were Bill Murray, Steve Fessler, and Mr. Wizard. He’s also mentioned in the song Walkin’ On the Sun by Smash Mouth. Were it possible for me to credit the many scientists that certainly owe their

I mention this because the legacy of Mr. Wizard was passed down. Don Herbert passed away on June 12, 2007. In an obituary published a few days later, Bill Nye writes that “Herbert’s techniques and performances helped create the United States’ first generation of homegrown rocket scientists just in time to respond to Sputnik. He sent us to the moon. He changed the world.” Reading the obituary, you cannot help but think that Bill Nye was also inspired to start his show by Mr. Wizard.

In fact, 20 years ago today, on 10 September 1993, the first episode of Bill Nye the Science Guy aired on PBS. It’s much more likely that readers of this blog have heard of Bill Nye; even though production of the show halted in 1998, PBS still airs reruns, and it’s commonly used in schools (did you know it won an incredible 19 Emmys?). I, for one, loved Bill Nye the Science Guy, and I still follow him to this day. I think it is impossible to narrow down the source of my initial interest in science, but I can certainly say that Bill Nye furthered my interest in science and experiments. He made science seem cool and powerful. To be clear, I know science is still cool and powerful, but I’m not so sure that’s the popular opinion. (As an aside: I also think math would really benefit from having our own Bill Nye).

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Twenty Mathematicians, Two Hard Problems, One Week, IdeaLab2013 Friday, Aug 2 2013 

July has been an exciting and busy month for me. I taught number theory 3 hours a day, 5 days a week, for 3 weeks to (mostly) devoted and motivated high school students in the Summer@Brown program. In the middle, I moved to Massachusetts. Immediately after the Summer@Brown program ended, I was given the opportunity to return to ICERM to participate in an experimental program called an IdeaLab.

IdeaLab invited 20 early career mathematicians to come together for a week and to generate ideas on two very different problems: Tipping Points in Climate Systems and Efficient Fully Homomorphic Encryption. Although I plan on writing a bit more about each of these problems and the IdeaLab process in action (at least from my point of view), I should say something about what these are.

Models of Earth’s climate are used all the time, to give daily weather reports, to predict and warn about hurricanes, to attempt to understand the effects of anthropogenic sources of carbon on long-term climate. As we know from uncertainty about weather reports, these models aren’t perfect. In particular, they don’t currently predict sudden, abrupt changes called ‘Tippling points.’ But are tipping points possible? There have been warm periods following ice-ages in the past, so it seems that there might be tipping points that aren’t modelled in the system. Understanding these form the basis for the idea behind the Tipping Points in Climate Systems project. This project also forms another link in Mathematics of Planet Earth.

On the other hand, homomorphic encryption is a topic in modern cryptography. To encrypt a message is to make it hard or impossible for others to read it unless they have a ‘key.’ You might think that you wouldn’t want someone holding onto an encrypted data to be able to do anything with the data, and in most modern encryption algorithms this is the case. But what if we were able to give Google an encrypted dataset and ask them to perform a search on it? Is it possible to have a secure encryption that would allow Google to do some sort of search algorithm and give us the results, but without Google ever understanding the data itself? It may seem far-fetched, but this is exactly the idea behind the Efficient Fully Homomorphic Encryption group. Surprisingly enough, it is possible. But known methods are obnoxiously slow and infeasible. This is why the group was after ‘efficient’ encryption.

So 20 early career mathematicians from all sorts of areas of mathematics gathered to think about these two questions. For the rest of this post, I’d like to talk about the structure and my thoughts on the IdeaLab process. In later posts, I’ll talk about each of the two major topics and what sorts of ideas came out of the process.

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Dancing ones PhD Saturday, Aug 11 2012 

In my dealings with the internet this week, I am reminded of a quote by William Arthur Ward, the professional inspirator:

We can throw stones, complain about them, stumble on them, climb over them, or build with them.

In particular, I have been notified by two different math-related things. Firstly, most importantly and more interestingly, my friend Diana Davis created a video entry for the “Dance your PhD” contest. It’s about Cutting Sequences on the Double Pentagon, and you can (and should) look at it on vimeo. It may even be the first math dance-your-PhD entry! You might even notice that I’m in the video, and am even waving madly (I had thought it surreptitious at the time) around 3:35.

That’s the positive one, the “Building with the Internet,” a creative use of the now-common-commodity. After the fold is the travesty.

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Ghostwritten Word Friday, Dec 2 2011 

I’ve just learned of the concept of ghostwriting, and I’m stunned.

A friend and fellow grad student of mine cannot believe that I’ve made it this far without imagining it to be possible. I asked around, and I realized that I was one of the few who wasn’t familiar with ghostwriting.

Before I go on, I should specify exactly what I mean. By ‘ghostwriting,’ I don’t mean situations where the President or another statesman gives a speech that they didn’t write themselves, but that was instead written by a ghostwriter. That makes a lot of sense to me. I refer to the cases where a student goes to a person or service, gives them their assignment, and pays for it to be completed. And by assignment, I don’t just mean 20 optimization problems in one variable calculus. I mean things like 20 page term papers on the parallels between the Meichi Revolution and American Occupation in Japan, or 50 page theses, or (so it’s claimed by some) doctoral dissertations.

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Reading Math Saturday, Oct 22 2011 

First, a recent gem from MathStackExchange:

Task: Calculate \displaystyle \sum_{i = 1}^{69} \sqrt{ \left( 1 + \frac{1}{i^2} + \frac{1}{(i+1)^2} \right) } as quickly as you can with pencil and paper only.

Yes, this is just another cute problem that turns out to have a very pleasant solution. Here’s how this one goes. (If you’re interested – try it out. There’s really only a few ways to proceed at first – so give it a whirl and any idea that has any promise will probably be the only idea with promise).

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From the Exchange Thursday, Jul 28 2011 

I speak of Math Stackexchange frequently for two reasons: because it is fantastically interesting and because I waste inordinate amounts of time on it. But I would like to again share some of the more interesting things from the exchange here.

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Giving Journals Saturday, Jul 23 2011 

Firstly, I wanted to note that keeping a frequently-updated blog is hard. It has its own set of challenges that need to be overcome. Bit by bit.

But today, I talk about a sort of funny experience. Suppose for a moment that you had acquired a set of low-level math journals throughout the undergrad days, journals like the College Mathematics Journal, Mathematics Magazine, etc. Presuming that you didn’t want to keep them in graduate school (I don’t – they’re heavy and I have online access), what would you do with them?

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Daily Math in Zagreb Tuesday, May 24 2011 

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.

Integration by Parts Thursday, May 12 2011 

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:

\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  \end{array}  … 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 \displaystyle \sum_{n\ge0} x^n/n! = e^x, but repeated integration by parts yields that \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 \displaystyle \int_0^1 \dfrac{ (-xlogx)^n}{n!} dx = (n+1)^{-(n+1)}. Summing over n, and remembering the Taylor expansion for e^x, one gets that \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 f and its inverse f^{-1}, we have that:
\displaystyle \int f(x)dx = xf(x) - \displaystyle \int xf'(x)dx =
= 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 f gives the integral of f^{-1} very cheaply, and this is why we use integration by parts to integrate things like lnx, arctanx, etc.  Similarly, one gets the reduction formulas necessary to integrate sin^n (x) or 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 cos(x)sin(x) or 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.’

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