Archive for October, 2008

One year down…

October 31, 2008

Since all the cool kids are doing it, allow me to post a “me too!” brief retrospective here to celebrate our first year.

Actually, “me too!” is pretty much how I got here to begin with.  I knew that two faculty at Nazareth College were starting a math blog, and after hearing about what they envisioned, I asked to be allowed in on the fun, even if I teach at the rival college up the road.  And fun it has been!  I haven’t had this much fun thinking, reading, and writing about vaguely-math-related-stuff since I was killing time in grad school on sci.math and other nether  realms of USENET, in the pre-web days of the net.  Thank you Batman and Ξ for sharing this space!

We (or at least I) don’t plan out topics in advance; most posts are either driven by stuff we see in the news, cool things that happen in class, or flashes of inspiration in the car driving home from work.  And there’s no telling what will hit a chord…  who knew that Godzilla would become a recurring character on a math blog?  or that Basketball Man and George Orwell Man would keep searching for those images?

Here’s to another year of friends near and far, food, horrid puns, and of course Godzilla, sharing both serious and whimsical slices of mathematics. Cheers!

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360 Days of 360

October 31, 2008

[Margin of error for this post: ±2%]

One year ago today, Ξ and I created this blog as an attempt to create even more work for ourselves. It began as an idle, I-don’t-want-to-work-right-now conversation while working (and not finishing) our newsletter, something along the lines of:

“It sure is a good thing we don’t have a blog, too, ’cause then we’d never get anything done.”

“Yeah…but if we did have one, what do you think we would call it?”

The rest, as they say, is history.

I started (blog) life as ‘360’, our blog admin, and posted a few times before creating my own identity: Batman. [You can ask my students where the name came from: them. One Calculus III student, to be specific. When introducing partial derivatives, I pointed out that the symbol ∂ doesn’t really have a name (\partial in LaTeX), so they could call it whatever they wanted. One of my quietest students said, “Batman,” the class went bananas, and I’ve used it ever since.] Now I post occasionally, rarely seriously, and lean heavily on Ξ to keep this blog afloat.

One thing I do less frequently now because I don’t have to: comment. If you were to look at our early comment log, you’d see a lot of Ξ, TwoPi, and Batman. Now, we have a (fairly) steady readership (thank you!), and other people are commenting for us! We had 951 hits in November of 2007. We average about that many hits every two days, now. I am incredibly pleased by this.

Things change over the course of a year, and some of my first posts were ambitious enough to claim that I would post regularly (Friday Software Reviews, anyone?). Right. My favorite post, though, has to be the jokes. It remains our most commented post of all time, and I certainly picked up a few new groaners to inflict upon my students.

One of the more surprising things I’ve found about us is that the most popular search term that leads you to 360 is [drumroll please]:

basketball

Yes, that’s correct: basketball. I wrote a post about March Madness back in…you know, and it contained this image:

We now get about 120-150 hits every weekday because of that. We like to think there’s one person responsible for this, whom we have affectionately named Basketball Guy. So here’s to you, Basketball Guy, a Real Man of Genius, and our most loyal reader. And here’s to all of you out there who make this blog worth writing. We couldn’t have succeeded without readers, and as long as people keep showing up to the party, we’ll keep playing the music.

Happy Anniversary!

October 31, 2008

One year ago today, Batman set up this site and wrote our very first post. It was seen by approximately 3 people. At that point I had a different username, but then I saw Batman and TwoPi’s names and I got jealous and picked the Greek symbol Xi (Ξ) because in grad school someone had told me once about a class where the prof was using Ξ for some variable, and it occurred in a fraction. The top was just Ξ, but the bottom was Ξ with a bar on top. The fraction ended up looking like this:

I thought that was about the coolest fraction I’d ever seen, and I would have been Xi-over-Xi-bar if I could have, but Xi was the closest I could come.

In thinking about the posts I’ve written, my artistic abilities are rather limited (as any of my students will tell you) so I’m disproportionately proud whenever I manage to create pictures for the posts. And it’s been a surprise (not any more, but at first) how often Godzilla makes guest appearances. He’s like Charo on the Love Boat: once he jumped into the blogging world (with a photo of himself eating Buffalo-Chicken Dip) he hasn’t looked back.

Happy Anniversary 360! And since the Anniversary falls on such an auspicious day, I’ll share one of my favorite sophomore-level math jokes:

Q: Why do mathematicians think that Halloween and Christmas fall on the same day?
A: Because 31 Oct = 25 Dec.*

* Think 31 Base 8 and 25 Base 10

Mathematical Halloween Costumes

October 30, 2008

Halloween is just a day away, and (unless you’re under the age of 15) you probably haven’t given much thought to what you might wear as a costume.

What is a mathematician to do?

I’ve been brainstorming for ideas for mathematically-themed Halloween costumes [sadly, the “storm” has been more of a light mist at best], but here are a few thoughts:

  • The Mod SquadDress up as your favorite character from the iconic 60’s television series, featuring three groovy cats fighting crime on the streets of LA.  [Short shameful confession:  the inspiration for this Halloween costume post was seeing Batman’s implication that we should teach science using rap, wondering what the sixties version of that might have been, and then remembering that I myself had dressed up as Pete Cochran of the Mod Squad on the day of my number theory final as an undergrad.  At the time I was puzzled that no one commented on the outfit, but then again, it was California, and many far stranger things happened on the streets and on campus to as little notice as possible.]
  • Or you could go political, choosing from a selection of “trig” puns:
    • Draw the first page of Trig Palin’s passport on a piece of cardboard and hang it around your neck. Voila! You’re a Trig Identity!
    • Dress up like a generic child. When asked, say you’re Sarah Palin’s youngest child’s doppelgänger. Yes, that’s right, a Trig Substitution.
    • sorry, uh, this isn’t a costume, but did you hear that if the McCain-Palin ticket wins the election, the Palin kids will be going to DC public schools? Yeah, that’s right. In a few years, it will lead to trig integration.  (“Thank you, thank you, I’m here all week… try the veal…”)
  • One could always dress up as your favorite mathematician.  (A trip to the costume store for powdered wigs if you’re thinking 17th C, or for ZZ-Top costumes if you’re thinking 19th C….)

So, what are you dressing up as for Halloween?

Frankenstein, Great Expectations, and Polygon

October 29, 2008

What’s the connection? Mary Shelley (born Mary Wollstonecraft Godwin) wrote Frankenstein; or, The Modern Promethius when she was a teenager, in 1818. The original Dr. Frankenstein’s monster didn’t look like the guy to the left: in the 3rd edition of the book (published in 1831) he looked like this:

So what does this have to do with Polygon? Well, Mary Shelley was born in The Polygon! The Polygon was here:

Some sites indicated that The Polygon was the name of the actual house, but after surfing the net when I really should have been grading doing some research I’m pretty sure that The Polygon was that immediate neighborhood, not one single building.

For example, in a book (Memoirs of the Author of a Vindication of the Rights of Woman) that her dad (William Godwin) wrote about her mom (Mary Wollstonecraft, who died 11 days after Mary was born), The Poygon is mentioned twice:

It is perhaps scarcely necessary to mention, that, influenced by the ideas I had long entertained upon the subject of cohabitation, I engaged an apartment, about twenty doors from our house in the Polygon, Somers Town, which I designed for the purpose of my study and literary occupations. Trifles however will be interesting to some readers, when they relate to the last period of the life of such a person as Mary. I will add therefore, that we were both of us of opinion, that it was possible for two persons to be too uniformly in each other’s society. Influenced by that opinion, it was my practice to repair to the apartment I have mentioned as soon as I rose, and frequently not to make my appearance in the Polygon, till the hour of dinner.

In digging around some more, I discovered someone else who lived in The Polygon: Charles Dickens! He wasn’t born there, but moved to 17 The Polygon, Somers Town in 1827 (more than a decade after Mary had left) at the tender age of seven when his family was evicted from their previous abode. [He only lived there about a year before moving.]

Finally, The Keeper of All that is Good and True says, “In 1784, the first housing was built at the “Polygon”, now the site of a council block of flats called “Oakshot Court”.” So I’m convinced that The Polygon is that neighborhood, maybe the plaza (which would likely be in the shape of a polygon). And the word Polygon is mathy, and Frankenstein is a pretty Halloweeny book, and Charles Dickens has some scary stuff it it (not monster-scary, but those debtors’ prisons don’t sound like much fun), so it all seems to fit the season.

A New Approach to Science?

October 28, 2008

Once again we turn to the writers at Robot Chicken to illuminate math and physics (also here if YouTube takes it down):

Lyrics Past the Jump

Bzzz Bzzz Bzzz: Bees can Count

October 27, 2008

It turns out that bees can count. Not very high (four is about as well as they can do), but still…they can COUNT!

My first question upon hearing this was, “How did they figure that out?” because I couldn’t really see an examiner sitting down with a bee to ask them to list a few numbers. It turns out (according to The Australian, found via the story on Yahoo! News) that Professor Mandyam Srinivasan placed landmarks in tunnels. If the bees were trained to go to the first landmark (where “trained” means that they discovered that nectar was always placed there), then they went to the first landmark to look for more nectar. If bees were trained to go to the second landmark, then that’s where they went to look for the treats. Same thing when the bees were trained to go to the third or fourth landmark.

But four was it: they couldn’t be trained to go to a fifth landmark. If there were more than four landmarks, they looked at all of them equally in their quest for sweet sweet nectar. (Note: it’s not clear to me if they were able to go to the first landmark if there were more than five, or if having more than five landmarks just completely confused them.)

Incidentally, a different news story in The Telegraph earlier this year announced that North American mosquito fish can also count to four. The fish could also distinguish between groups of very different sizes (16 versus 8, but not 16 versus 12). No word on how bees would do at the dots test.

One, Two, Three, Four, Six [Again. And then again!]

October 26, 2008

I’m teaching a math course for non-majors, and right now we’re talking about Induction versus Deduction. I have some neat examples of Induction, like the fact that the US presidents elected in 1840, 1860, 1880, 1900, 1920, 1940, and 1960 all died in office, but Ronald Reagan did not. I’ve found, though, that these cultural examples don’t carry as much weight for the students as a mathematical pattern that continues for a while and then stops. Hence my interest in Patterns that Fail.

In this vein, last week I looked at how many necklaces could be made out of N beads, where the beads could be two different colors, and it turns out that the number of necklaces follows the pattern one [for 0 beads], two [for 1 bead], three [for 2 beads], four [for 3 beads], and then six [for 4 beads]. But there’s another setup that gives the same pattern 1, 2, 3, 4 before jumping to 6. This setup involves covering 3xN rectangles with dominoes that are 1×3 or 3×1 (tri-ominoes? But I think those are L-shaped).

If you start with N=2 (to avoid the sequence beginning 1,1,…), there is one way:

If N=3, there are two ways:

If N=4, there are three ways:

If N=5, there are four ways:

But if N=6, suddenlythere are six ways!

After that, the pattern grows in larger steps [following the recursive pattern a(n)=a(n-1)+a(n-3)].

Incidentally, there’s another pattern that starts off 1, 2, 3, 4, 6, ….: the number of ways to make N cents in 1¢, 2¢, 3¢, 5¢, 10¢, 20¢, 25¢, 50¢ and/or 100¢ coins, all of which are or have been valid US coins. For example:

  • 1¢ can only be made with a 1¢ coin. [1 way]
  • 2¢ can be made with two 1¢ coins or 1 2¢ coin. [2 ways]
  • 3¢ can be made with three 1¢ coins, a 1¢ and a 2¢ coin, and a 3¢ coin. [3 ways]
  • 4¢ can be made with four 1¢ coins, two 1¢ and one 2¢ coins, one 1¢ and one 3¢ coins, or two 2¢ coins. [4 ways]
  • 5¢ an be made with five 1¢ coins, three 1¢ coins and one 2¢ coin, two 1¢ coin and one 3¢ coins, one 1¢ coin and two 2¢ coins, one 3¢ coin and two one 2¢ coins, or one 5¢ coin. [6 ways]

This sequence continues 8, 10, 13, 16…so it’s different than the previous sequences, giving me lots of examples to choose from in class!

Predicting Avalanches and Tsunamis

October 25, 2008

Equations, exciting and new
Oh Tsunamis, they’re expecting you
And Math, life’s sweetest reward
Let it float, it floats back to you

Savage-Hutter
soon will be making another run
With Coulomb
promises something for everyone
In the course of a landslide
On Alborón can still be romance

Because Tsunamis won’t hurt anymore
There are open smiles on these friendly shores
It’s Math
Welcome aboard it’s math
Welcome aboard it’s math

Thank you, thank you, we’ll be here all week. Today’s news story was brought to you by Science Direct, and concerns a paper written by E.D. Fernández-Nieto, F. Bouchut, D. Bresch, M.J. Castro Díaz, and A. Mangeney. They took the Savage-Hutter equations, which have already been used for rock avalanches, added some information about the Coulomb friction term (which is related to the fact that when a liquid spills it spreads evenly all over the floor, but when something more solid like sand spills it forms a pile), and used it to examine landslides from the Spanish island of Alborón (Almería). It might even be possible to predict tsunamis, although not necessarily to prevent them.

And if you can’t concentrate on any of that because you have the theme song for The Love Boat stuck in your head, here’s a little something TwoPi and I found when we did a guest post for Our Best Friend Craig on Puntabulous when he was on a cruise:

Carnival of Mathematics #42

October 24, 2008

It’s time for the answer to life, the universe, and everything in Carnival Form:  Carnival of Mathematics #42 is up over at The Endeavour, a blog by John D. Cook.    His blog has neat articles like how to convert files to .pdf and Jenga Math (weakening the hypothesis of a theorem without causing the whole thing to collapse).   The Carnival has lots of neat articles on logic, computer science, and how the check-sum digit works on credit cards.  So pull up a chair and Enjoy!

Integration by parts via iterated integrals

October 22, 2008

(Another installment of an occasional series, featuring obfuscated proofs of elementary results.  Previously: Using calculus to generate the quadratic formula.  )

When faced with a challenging integration problem, sometimes some ingenuity is called for.  What follows is a technique that I vividly remember teaching in the late 1980s in a multivariable calculus course, which probably means this can be found in the 5th or 6th edition of Thomas and Finney.  (I don’t have a copy of that book anymore, so I can’t readily check.)

We’ll start with an example:  find \int_{1}^4 \ln (x) \; dx.

If you don’t know the antiderivative of the natural logarithm function, what to do?  Well, if you know its derivative, you might replace \ln(x) with \int_1^x \frac1t \; dt.  If we make that substitution, we arrive at the double integral \int_1^4 \; \int_1^x \frac{1}{t} \; dt \; dx.

Next, we interchange the order of integration.

Graphing the region in the x,t plane, we see that the double integral can be rewritten as:

\int_1^4 \; \int_1^x \; \frac1t \; dt \; dx \; = \; \int_1^4 \int_t^4 \frac1t \; dx \; dt

This works out well, as now the inner antiderivative is simple.  We get

\int_1^4 \int_t^4 \frac{1}{t} \; dx \; dt \; = \; \int_1^4 \frac{4-t}{t} \; dt

which simplifies to

\int_1^4 \frac{4}{t} - 1 \; dt = ( 4 \ln(4) - 4 ) - (4 \ln(1) - 1) = 4 \ln(4) - 3.

I remember doing several examples like this for my students (as they’d been assigned as homework exercises), and thought it rather odd that every example in the text was one that could just as easily have been done using integration by parts.

And indeed, one can use this interated integral argument in place of integration by parts for any definite integral:

Consider \int_a^b f(x) g'(x) \; dx.  If we replace f(x) by \int_{f^{-1}(0)}^x f'(t) \; dt, we find that

\int_a^b f(x) g'(x) \; dx = \int_a^b \int_{f^{-1}(0)}^x \; f'(t) g'(x) \; dt \; dx

Interchanging the order of integration is a bit tricky:

In general we end up with the sum of two iterated integrals:

\int_{f^{-1}(0)}^a \int_a^b f'(t) g'(x) \;dx\;dt + \int_a^b \int_t^b f'(t) g'(x) \; dx\; dt

The first integral yields

\int_{f^{-1}(0)}^a f'(t) (g(b)-g(a)) \; dt = f(a)(g(b)-g(a)).

The second integral yields

\int_a^b f'(t) (g(b) - g(t)) \; dt = g(b)(f(b)-f(a)) - \int_a^b f'(t) g(t) dt.

Adding these two answers gives the final result:

\int_a^b f(x) g'(x) dx = f(b)g(b) - f(a)g(a) - \int_a^b f'(t) g(t) dt

which is the same as the integration by parts formula:

\int_a^b f(x) g'(x) dx = \left. f(x)g(x) \right|_a^b - \int_a^b f'(x) g(x) dx

The Fall Newsletter: A long name, some money, and some math

October 21, 2008

The Fall 2008 issue of the Nazareth College Math Department Newsletter has just been posted! Each Newsletter is named after a different mathematician, and this one is called Le Tonnelier de Breteuil Marquise du Châtelet Gazette after Gabrielle Émilie Le Tonnelier de Breteuil Marquise du Châtelet, the 18th century mathematician who (among other things) translated Newton’s Principia into French. [Question: How much fun did we have coming up with the title?]

The feature article was written by one of our juniors about her study-abroad experience in Germany last fall, and many of the other articles first made their appearance here, but one of the inside stories might have a wider audience — namely, people thinking about college or grad school — and so might shameless plug be worth a special mention shameless plug. There are three scholarship opportunities for math and science folk: two of them (just recently funded by the National Science Foundation! Hooray!) are local to Nazareth College, but the third is available throughout New York State, and might have parallels in other states as well.

  • The Robert Noyce Scholarship at Naz is aimed at undergraduate and graduate students seeking teaching certification at the adolescent or childhood/middle childhood levels in science or math (in exchange for service in a “high need” district).
  • The Science and Mathematics Scholarship Program at Naz offers scholarships up to $10,000 for “promising, financially needy students” enrolling in biology, chemistry, or mathematics majors.
  • Finally, the New York State Math and Science Teaching Incentive Scholarships provide awards to undergraduate or graduate students pursuing careers as secondary math and science teachers, in exchange for five years of full-time employment in the state.

Truth be told, although Batman and I work at Naz (and are the Executive Editors of the Newsletter, in case that wasn’t quite blatant enough), TwoPi actually works at St. John Fisher College just down the road. And they, too, are offering money:

  • The Science Scholars Program at SJFC offers $12,000 scholarship for math, computer science, and science majors (entering as freshmen right out of high school).

If scholarships aren’t what you’re looking for, you can still check out the back page of the newsletter for some fun math problems to play around with. My current favorite of the bunch is:

Suppose the number N satisfies log2(log3(log5(log7 N))) = 4.
How many different prime factors does N have?

Math Mistake costs several hundred jobs

October 20, 2008

CNN reported on Friday (and JD2718 mentioned on Saturday) that the Dallas school system had to lay off some employees after realizing that its budget was just a tad short, where “just a tad” stands for $84,000,000 [more even than the $64 million first reported]. Whoops! According to the CNN news article:

The district laid off 375 teachers and 40 counselors and assistant principals Thursday, and transferred 460 teachers to other schools within the district.

The deficit was caused by a massive miscalculation in the budget, CNN affiliate WFAA-TV reported.

So that got me wondering what possible miscalculation could have led to that kind of error. It turns out that a large part of it was due to underestimating the average salary of the teachers (which runs on the order of $50,000 for elementary school teachers, according to this site). In this September 23 video, Dr. Michael Hinojosa (superintendent of the Dallas Independent School District) stated:

When you take, you underestimate your average teacher’s salary by $3900 and you multiply that over 11,000 teachers, then that creates a huge budget error.

Yes. Yes it does. About 43 million dollars. The rest of the error was apparently due to not following a formula for how many administrative positions to have at each of the 225 DISD campuses, resulting in 1-2 “extra” people at each campus, for a total of 338 extra positions.

Moral: Being close is not always good enough.

Using calculus to generate the quadratic formula

October 19, 2008

I get (unreasonable considerable) joy in finding fancy new proofs of elementary results, proofs that might come under the heading of mathematics made difficult or obscure. (I’ve never seen Linderholm’s book, but suspect it would appeal to a twisted part of my psyche.) I don’t quite understand the psychology behind this liking, but there it is nonetheless.

One of my favorite examples follows: solving quadratic equations using integration to complete the square.

Suppose f(x)=ax^2+bx+c, and we want to find the solutions to f(x)=0.

Note that f'(x) = 2ax+b and f(0) = c, and thus it must be that f(x) = c+ \int_0^x 2at+b \; dt .

We compute this antiderivative using the change of variables w=2at+b, \quad dw = 2a\;dt, which leads to c + \int_b^{2x+b} \frac1{2a} \; w \; dw. This last expression is equal to c + \left( \frac1{4a} (2ax+b)^2 - \frac{b^2}{4a} \right).

Thus the roots of f(x) = 0 are found by solving the equation c + \frac{ (2ax+b)^2 - b^2 }{4a} = 0.

This leads to (2ax+b)^2 -b^2 = -4ac, and thus 2ax+b = \pm \sqrt{b^2 - 4ac}, and so finally we arrive at the roots x = \frac{ -b \pm \sqrt{b^2 - 4ac}}{2a} .

Forthcoming: Using Fubini’s Theorem to prove Integration by Parts

Blogging on the YMN

October 17, 2008

I started reading the *Concerns of Young Mathematicians* from the YMN (Young Mathematicians’ Network) back in the early 1990s, back when it was a weekly email newsletter. The email has been replaced by a web site, and this month introduces two new folk who will be contributing regularly:

  • Robert Talbert, who is also the author of the blog Casting out Nines. His first YMN article is “The hiring process as risk management“. I never viewed the hiring process in quite those terms, but he makes some really interesting points.
  • An anonymous faculty member who is chairing a search committee. Their series, “Blogging our search“, gives some of the inside scoop on the search process.

[This makes it sound like there’s only info about the job search; I believe that others will be joining them to talk about additional subjects of interests to Young Mathematicians, whether undergrads, graduate students, working folk, or recreational mathematicians.]

*In the interests of full disclosure, I’m on the editorial board of the YMN. And I’m totally excited about these new columnists!