Monthly Archives: May 2013

teaching

A smooth but not analytic function


In the current assignment for real analysis, we consider the following function
\[
f(x) =
\begin{cases}
e^{-1/x^2} & x \neq 0\\
0 & x = 0.
\end{cases}
\]
We are asked to show that \(f^{(n)}(0) = 0\) for all \(n\), and that the Taylor series for \(f\) at \(0\) converges to \(f(x)\) only for \(x \neq 0\).

To compute the derivatives \(f^{(n)}(0)\), it is easiest to use induction on \(n\). For \(n = 1\),
\[
f'(0) = \lim_{x \to 0} \frac{f(x) – f(0)}{x – 0} = \lim_{x \to 0} \frac 1 x e^{-1 / x^2}.
\]
To compute the limit, we first use the change of variables \(y = 1 / x\) so that as \(y \to \pm \infty\) we have \(x \to 0\). Then
\[
\lim_{x \to 0} \frac 1 x e^{-1 / x^2} = \lim_{y \to \pm \infty} \frac{y}{e^{y^2}}.
\]
Using L’Hopital’s rule, it is easy to verify that the latter limit is zero. In fact, a similar argument shows that for any \(k\)
\[
\lim_{x \to 0} \frac{1}{x^k} e^{-1 / x^2} = 0.
\]
We will need this fact later on. At any rate, we’ve shown that \(f'(0) = 0\).

For the inductive step, assume that \(f^{(n)}(0) = 0\). Then
\[
f^{(n + 1)}(0) = \lim_{x \to 0} \frac{f^{(n)}(x) – f^{(n)}(0)}{x – 0} = \lim_{x \to 0} \frac{1}{x} f^{(n)}(x).
\]
After computing the first few derivatives, \(f'(x),\ f”(x), \ldots\) for \(x \neq 0\), you should be convinced that \(f'(x)\) is of the form
\[
f^{(n)}(x) = \left (\frac{a_k}{x^k} + \frac{a_{k-1}}{x^{k-1}} + \cdots + a_0 \right ) e^{-1 / x^2}.
\]
Using the same trick as before, we can compute
\[
\lim_{x \to 0} \frac{a_i}{x^{i+1}} e^{-1 / x^2} = 0,
\]
hence
\[
f^{(n + 1)}(0) = \lim_{x \to 0} \frac{1}{x} f^{(n)}(x) = 0.
\]

Since all of the derivatives of \(f\) are \(0\) at \(x = 0\), the Taylor series there is
\[
\sum_{k = 0}^\infty f^{(k)}(0) x^k = 0.
\]
However, \(f(x) \neq 0\) for \(x \neq 0\), so the Taylor series only agrees with \(f\) at \(x = 0\).

brewing

Workman’s Friend Porter

Brewing is alchemy. It is equal parts art and science, steeped in history and suffused in community. For me, brewing is almost more about the process than the finished product. When I brew, I feel I am participating in a dialog that spans scores of generations through the millennia. Of course, the particular techniques I apply in my kitchen are a far cry from the methods of my ancestors, but the basics remain the same. Steep grain. Boil wart. Ferment. Enjoy with friends.

This holiday weekend, I finally have a chance to brew another batch of beer. With Alivia’s encouragement, I decided to make a porter. I’ve spent some time doing my homework on this style of ale to craft a recipe. I found the chapter on porters in Daniels’ Designing Great Beers most helpful. In the grain bill, I am trying to recreate some aspects of the brown malt that was the backbone of porters in the 18th and 19th centuries. Brown malt fell out of favor after the 19th century because of its low yield compared to the combination of pale and roasted malts used in most dark ales today. I am hoping that a combination of pale, caramel, Munich and smoked malts will imbue my brew some of the characteristics of an 18th century British porter.

To contrast the faux-traditional malt profile, I am using a decidedly contemporary west coast hop profile. Inspired by Rogue’s Mocha Porter, I am using American Perle hops for bittering and Sterling for flavor and aroma. Here is the recipe for a 5.5 gallon batch:

Grains

  • 8 lb pale malt (2 row)
  • 1 lb caramel 60
  • 1 lb Munich
  • 8 oz chocolate
  • 3 oz black malt
  • 1.5 oz peat smoked malt

Hops

  • 1.5 oz American Perle (60 min)
  • 1 oz Sterling (30 min)
  • 1 oz Sterling (5 min)

Yeast

  • British ale (Wyeast 1098)

I plan on doing a single temperature infusion mash at 152 degrees. If all goes according to plan, I should end up with an original gravity of 1.053 and bitterness of around 45 IBU. I am anticipating a final gravity of around 1.014 for an ABV of 5.1%.

I decided to name this beer The Workman’s Friend after Flann O’Brien’s poem of the same name:

When things go wrong and will not come right,
Though you do the best you can,
When life looks black as the hour of night –
A pint of plain is your only man.

When money’s tight and hard to get
And your horse has also ran,
When all you have is a heap of debt –
A pint of plain is your only man.

When health is bad and your heart feels strange,
And your face is pale and wan,
When doctors say you need a change,
A pint of plain is your only man.

When food is scarce and your larder bare
And no rashers grease your pan,
When hunger grows as your meals are rare –
A pint of plain is your only man.

In time of trouble and lousey strife,
You have still got a darlint plan
You still can turn to a brighter life –
A pint of plain is your only man.

— Flann O’Brien

teaching

Math 32A Midterm II Review

I’ve written up some review questions for the second midterm for Math 32A this quarter. Here is a link to the questions. Here are the solutions. As usual, let me know in the comments below if you find any mistakes or if anything is unclear.

computing

Make a Centralized Bibliographic Database with BibTeX!

Like most working mathematicians I know, I use LaTeX for all of my mathematical typesetting. Until recently, the vast majority of my writing was for assignments for classes and personal essays where I didn’t need to cite any sources. Fortunately there is a wonderful tool called BibTeX that easily manages citations for when inevitably I need to cite others’ work. I have started compiling a centralized database of all the papers and books that I cite in my own writings. All of the sources are in a single file (which I called sources.bib). In order for BibTeX to find sources.bib whenever I reference it, I placed the file in my local texmf tree:

.../texmf/bibtex/bib/sources.bib

Now any time I feel the need to cite, for example, Cover and Thomas’ Elements of Information Theory, all I need to do is put

\cite{CT06}

in my .tex file and BibTeX takes care of the rest.

 

computer science edutainment expository math

Information Theory

I have just uploaded the beginnings of an essay on information theory to my website. You can see the essay (in its currently incomplete state) here. Information theory was first described by Claude Shannon in his groundbreaking 1948 paper, A Mathematical Theory of Communication. What is particularly surprising about Shannon’s truly remarkable paper is its completeness. Not only does Shannon suggest a mathematical model for (digital) communication and information, but he produces a huge array of fundamental results for his model. It is exceptionally rare that such a complete theory comes into being so fully formed, like the birth of Athena. A decade ago, UCSD produced documentary about Shannon’s life and work, which I have posted below.

math musings

The Paradox of the Proof

The Paradox of the Proof is an essay by Caroline Chen about Shinichi Mochizuki‘s supposed proof of the abc conjecture. The article is accessible and very well written. I especially like that the article touches on the social aspect of mathematics. It seems a common conception about math that mathematical statements are either true or false, and that the truth or falsity is somehow “objective.” However, Mochizuki’s work shows that this is not the complete story. Mochizuki claims to have a proof of the abc conjecture, but literally no one (except presumably Mochizuki himself) understands the proof. So the truth of the abc conjecture still seems in limbo. This gives credence to my belief that a mathematical proof is not merely a mechanical verification that a statement is true. Rather, a proof is a narrative about why a statement is true — if there is no consensus in the mathematical community that a proof is valid, it may as well not exist. Chen’s article does a wonderful job of conveying this tension between the austere and rigorous abstractions of mathematics and its social nature that gives the field life.