The Unapologetic Mathematician

ARML Scrimmage Power Question

I helped the Howard County and Baltimore County ARML teams practice tonight by joining the group of local citizens and team alumni to field a scrimmage team. As usual, my favorite part is the power question. It follows, as printed, but less the (unnecessary) diagrams:

Consider the function

$\displaystyle\phi(t)=\left(\frac{2t}{t^2+1},\frac{t^2-1}{t^2+1}\right)=(x,y)$

which maps the real number $t$ to the a coordinate in the $x$$y$ plane. Assume throughout that $q$, $r$, $s$, $t$, and $u$ are real numbers.

(1) Compute $\phi(1)$, $\phi(1/2)$, $\phi(2)$, $\phi(-1)$, $\phi(-1/2)$, and $\phi(-2)$. Sketch a plot of these points, superimposed on the unit circle.

(2) Show that $\phi$ is one-to-one. That is, show that if $\phi(s)=\phi(t)$, then $s=t$.

(3) Let $(x_\phi,y_\phi)$ be the intersection point between the unit circle and the line connecting $(0,1)$ and $(t,0)$. Prove that $\phi(t)=(x_\phi,y_\phi)$.

(4) Show that $(x,y)$ is an ordered pair of rational numbers on the unit circle different from $(0,1)$ if and only if there is a rational number $t$ such that $\phi(t)=(x,y)$. (This result allows us to deduce that there are infinitely (countably) many rational points on the unit circle.)

According to problem 3, $\phi(t)$ is a particular geometric mapping of a single point on the real line to the unit circle. Now, we will be concerned with the relationship between the pairs of points, which will lead to a way of doing arithmetic by geometry. Use these definitions:

• Let $\left\{\phi(s),\phi(t)\right\}$ be a “vertical pair” if either $s=t=1$, or $s=t=-1$, or $st\neq0$ and $\phi(s) and$latex \phi(t)\$ are two different points on the same vertical line.
• Let $\left\{\phi(s),\phi(t)\right\}$ be a “horizontal pair” if either $s=t=0$, or $\phi(s)$ and $\phi(t)$ are two different points on the same horizontal line.
• Let $\left\{\phi(s),\phi(t)\right\}$ be a “diametric pair” if $\phi(s)$ and $\phi(t)$ are two different end points of the same diameter of the circle.

(5) (a) Prove that for all $s$ and $t$, $\left\{\phi(s),\phi(t)\right\}$ is a vertical pair if and only if $st=1$.
(b) Prove that for all $s$ and $t$, $\left\{\phi(s),\phi(t)\right\}$ is a horizontal pair if and only if $s=-t$.
(c) Determine and prove a relationship between $s$ and $t$ that is a necessary and sufficient condition for $\left\{\phi(s),\phi(t)\right\}$ to be a diametric pair.

(6) (a) Suppose that $\left\{\phi(s),\phi(t)\right\}$ is not a vertical pair. Then, the straight line through them (if $\phi(s)=\phi(t)$, use the tangent line to the circle at that point) intersects the $y$-axis at the point $(0,b)$. Find $b$ in terms of $s$ and $t$, and simplify and prove your answer.
(b) Draw the straight line through the point $(1,0)$ and $(0,b)$, where $(0,b)$ is the point described in problem (5a). Let $\phi(u)$ denote the point of intersection of this line and the circle. Prove that $u=st$.

(7) (a) Suppose that $\left\{\phi(s),\phi(t)\right\}$ is not a horizontal pair. Then, the straight line through them (if $\phi(s)=\phi(t)$, use the tangent line to the circle at that point) intersects the horizontal line $y=1$ at the point $(a,1)$. Find $a$ in terms of $s$ and $t$, and simplify and prove your answer.
(b) Draw the straight line through the point $(0,-1)$ and $(a,1)$, where $(a,1)$ is the point described in problem (6a). Let $\phi(u)$ denote the point of intersection of this line and the circle. Prove that $u=s+t$.

(8) Suppose $q$, $r$, $s$, and $t$ are distinct real numbers such that $qr=st$ and such that the line containing $\phi(q)$ and $\phi(s)$ intersects the line containing $\phi(r)$ and $\phi(t)$. Find the $y$-coordinate of the intersection point in terms of $s$ and $t$ only.

(9) Let $s$ and $t$ be distinct real numbers such that $st>0$. Given only the unit circle, the $x$– and $y$– axes, the points $\phi(s)$ and $\phi(t)$, and a straitedge (but no compass), determine a method to construct the point $\phi(\sqrt{st})$ that uses no more than $5$ line segments. Prove why the construction works and provide a sketch.

(10) Given only the unit circle, the $x$-and $y$– axes, the point $(1,1)$, and a straightedge (but no compass), describe a method to construct the point $\left(-\frac{2\sqrt{3}}{3},0\right)$.

May 21, 2009 Posted by | Uncategorized | 2 Comments

Happy New Year

I’m a little under two hours late, but here’s to all who celebrate today, including the contingent of grad students I knew back at Yale.

And remember: Nowruz is good ruz.

Joint Meetings 2009 — Day 4

Sorry for not posting yesterday, but I was sort of run-down. I”m taking today as a break before I join Knots in Washington tomorrow, already in progress. Anyhow, the last day of the Joint Meetings is always slow, and there’s nothing much for me to talk about here. Instead, I’ll put up some pictures.

Joint Meetings 2009 — Day 3

Today’s special session on homotopy theory and higher categories seems to have pushed its higher categories until the afternoon, so I got a chance to see Dan Teague (of Dartmouth) talk about “Making Math out of Style”.

This was rather interesting to me, since the jumping-off point was the identification of Pollack paintings by box-counting dimensions. I really liked this story when it came out, since it’s a great story to relate mathematics to art. The talk continued to discuss efforts to identify authorship of some of the Federalist Papers, and of one of the Wizard of Oz books. Then there was identifying forged Van Gogh paintings (which I think I saw on Scientific American Frontiers a few months ago). Neat stuff.

In the afternoon, John Baez led off, talking about the classifying space of a 2-group. I’ll also him later discussing groupoidification in the categorification and link homology. I wanted to make this post now in a bit of down time so I could remind people that I’ll be at Tryst at 8, and to pass on this bit of wisdom from Baez’ first talk: $X$ is just $M$ in a really weird font.

[UPDATE]: Paul is right in his comment below. Dan Rockmore gave the talk, and Dan Teague introduced him. I met someone else the next day who confirmed both this fact and that he was initially confused by it as well.

January 7, 2009 Posted by | Uncategorized | 4 Comments

Joint Meetings 2009 — day 2

Today was a little thin. I had to meet someone in the exhibit hall when it opened, and the talks I saw after that in the morning were sort of lackluster. After lunch I saw a couple topology and applied mathematics talks, but then had to head off for another meeting about a job prospect. After that rather than sticking around for Mikhail Khovanov’s invited address (I probably know what he’d say anyway), I decided to hit the metro and try to beat the beltway traffic.

One talk in the early morning caught my attention. David Clark talked about the functoriality of the $\mathfrak{sl}_3$ analogue of Khovanov homology (which is based around $\mathfrak{sl}_2$. The talk itself I don’t care to talk about much here, but I was glad to see that the first step was to pass from links to tangles, and to treat them as the natural setting. Now if I can just get the term “tangle covariant” to catch on…

Oh, I wasn’t able to join the Secret Blogging Seminar’s drinking tonight. Tomorrow night, however, I’m thinking I’ll be at Tryst. I’m done with special sessions by 6, and I’ll be wanting to have dinner of course. So I’ll say “8 PM”. Here’s a map.

Joint Meetings 2009 — day 1

Well, I made some good non-academic contacts already. I’d rather not go into details, since being too talky might be a problem when prospective employers look to Google and find me as the top hit for my name and subject. But I’m feeling good about my prospects, even without looking at the wilds of federal government and contracting jobs.

Anyhow, as for mathematics there were a number of good talks, but most of them were either what was expected from the speaker, or felt pretty technical. One, though, really grabbed me. Kerry Luse, formerly a student of Yongwu Rong’s at George Washington, spoke about “A transition polynomial for signed Feynman diagrams”. She started with the chord diagram of a knot and added a sign to each chord. If you read the signs as orientations, you get Feynman diagrams for a single species of noninteracting, non-self-dual particles. Alternately, you can interpret the diagrams as arising from RNA secondary structures, as she did. Either way, she was looking for a polynomial invariant to be calculated from such a diagram, and she came up with some really interesting results from her choice. One property in particular was the fact that the resulting polynomial (as applied to chord diagrams arising from knots) is multiplicative under connected sums of links. This makes me think it’s got something to do with the Alexander polynomial.

She also mentioned chord diagrams for links, with more than one loop, which I don’t think I’ve ever considered as such. Immediately this made me think of extending to tangles (naturally), and then that these chord diagrams may themselves form a category of their own. Is there some sort of duality here? If so it might turn connected sum on one side into disjoint union on the other side, which could provide a fascinating connection between classical and quantum topology…

See, I’m not going to be stopping research, and definitely not stopping this project here (thanks, btw, for the comments), but I just need to get out of the academic game I’ve been playing the last few years.

Anyhow, since it’s been weeks since I’ve been at home cooking for myself (thanks to Dad insisting on doing it all), I figured I’d have a bonus “I (Didn’t) Made It!”:

Eating at the Afghan Grill

At the Afghan Grill, just around the corner from the Marriott, I’m having the mantoo, and across the table from me is the lamb qabili palao. So who ordered the Afghan equivalent of biryani?

This guy! I also ran into Jesse Johnson this morning. Oh, and if Sarah from John Hopkins is reading this and is at the meetings, she really needs to drop me an email so she can join the fun.

January 6, 2009 Posted by | Uncategorized | 1 Comment

Math and Philosophy

An aspiring philosopher of mathematics got hold of my post on categorification, and it’s leading to (what I find to be) an interesting discussion of how Hume’s Principle fits into the story, and just what the Peano axioms (not to mention “the natural numbers”) mean. Follow along (or jump in!) at if-then knots — an excellent title, in my opinion.

Fall Break

WKU is off for a few days, so I’ll take a break too. I should be back Monday.

Today is my last long drive for a while. I’ll try to post a Sunday Sample tonight, or maybe tomorrow. But for now, here’s a problem to chew on.

In a multiple choice test, one question was illegible, but the following choice of answers was clearly printed. Which must be the right answer?

1. All of the below
2. None of the below
3. All of the above
4. One of the above
5. None of the above
6. None of the above

August 24, 2008 Posted by | Uncategorized | 15 Comments

Testing…

While I wait for my food to cool a little bit, I thought I’d try out this WordPress iPhone app. Kick the e-tires and so on.

I’m not going to make a one-day run this time, since I didn’t leave New Orleans until 13:00 CDT. So finally I have the excuse to stop in Dayton, TN tonight. Tomorrow morning I’ll see the Scopes Trial museum at 08:00 and be on my way. Yes, there will be pictures.

August 1, 2008 Posted by | Uncategorized | 1 Comment