Mathematics
jrosenhouse | August 23, 2011From the department of self-promotion, let me call attention to the current volume of The Electronic Journal of Combinatorics. If you click on the link and scroll down to P164, you will find a barn-burning, rhetorical masterpiece of a paper entitled, “Isoperimetric Numbers of Regular Graphs of High Degree With Applications to Arithmetic Riemann Surfaces.” And who wrote this paper, you ask, the suspense practically killing you? That would be my long time friend and collaborator Dominic Lanphier, of Western Kentucky University, along with yours truly. Yay! Always nice to have one more line…
jrosenhouse | August 22, 2011One upside to my recent convalescence has been that I have had plenty of time for reading. Currently I'm working my way through Graham Oppy's book Philosophical Perspectives on Infinity, published by Cambridge University Press in 2006. Oppy is best known as a philosopher of religion, writing from a generally atheistic perspective. His book Arguing About Gods is really excellent, thought it definitely does not make for light reading.
As for the present book, I'm only through the first two chapters so far. I think I'm going to like the rest, though, since the preface contains remarks like…
jrosenhouse | August 16, 2011A number of years ago I attended an ID conference near Kansas City. One of the breakout sessions featured a fellow from the Heritage Foundation (an ultra-right-wing political think tank) presenting a lecture about probability and evolution. His talk was mostly just a watered-down version of William Dembski's standard talking points. His triumphant conclusion was that the probability of something or other having evolved by natural processes was one over something enormous, from which he concluded that evolution had been refuted.
There were only about fifteen of us in the room. During the…
jrosenhouse | July 18, 2011A while back I I began a discussion about deriving formulas for solving polynomial equations. We saw that linear and quadratic polynomials did not pose much of a challenge. But cubic polynomials are considerably more complex. The set-up was that we had a polynomial equation of this form:
\[
x^3+ax^2+bx+c=0
\]
We can assume that the leading coefficient is one, so long as we're OK with the idea that the other coefficients might be fractions. If we have a cubic polynomial whose leading coefficient is not one, then we can simply divide through by whatever it is.
We got as far as saying…
sciencepunk | July 18, 2011YouTube artist and multi-instrumentalist Michael John Blake has been exploring the musical patterns within mathematics, by assigning each note on the major scale a numeral. In this wonderful composition he plays Tau, a number used in a similar way to Pi, for calculating values of circles and whatnot.
jrosenhouse | June 20, 2011We mathematician types like solving polynomial equations. The simplest such equations are the linear ones, meaning that the variable appears to the exponent one. They have the general form:
\[
ax+b=0.
\]
If you remember anything at all from your basic algebra classes, then you know that this is readily solved by bringing the b to the other side and dividing by a. We obtain
\[
x=\frac{-b}{a}.
\]
Of course, we are assuming here that a is not zero, but let's not be overly pedantic. We can think of this as “the linear formula,” since it can be used to solve any linear equation we might…
mikethemadbiologist | June 2, 2011...don't understand fractions. A recent Gallup poll asked people what percentage of Americans were gay and lesbian. The results? 52 percent estimated that twenty percent or more of the population is gay or lesbian:
Keep in mind that most estimates put the lesbian and gay population at around three percent. While the Gallup pollsters and other commentators ascribe this to increasing acceptance, I think it's something more basic--a lot of people don't really instinctively make the link between fractions and percentages:
The arithmetic gap is the most obvious one: profs over a certain age (…
jrosenhouse | May 23, 2011Jerry Coyne calls our attention to this abstract, from a recent issue of Proceedings of the National Academy of Sciences:
We show how to measure the failure of the Whitney move in dimension 4 by constructing higher-order intersection invariants of Whitney towers built from iterated Whitney disks on immersed surfaces in 4-manifolds. For Whitney towers on immersed disks in the 4-ball, we identify some of these new invariants with previously known link invariants such as Milnor, Sato-Levine, and Arf invariants. We also define higher-order Sato-Levine and Arf invariants and show that these…
jrosenhouse | May 9, 2011In last week's post we discussed perfect numbers. These were numbers, like 6, 28 and 496, that are equal to the sums of their proper divisors. We referred to Euler's formula, which claims that every even perfect number has the form
\[
2^{p-1} \left(2^p-1 \right),
\]
where the term in parentheses is prime. As we discussed last week, the term in parentheses is known as a Mersenne prime, which entails that the exponent p is prime as well.
Our goal this week is to prove this formula. This is a very beautiful proof, in my view. It has tremendous flow, by which I mean that there is no one…
jrosenhouse | May 2, 2011In last week's post, we discussed Mersenne primes. These were primes of the form:
\[
2^p-1, \phantom{x} \textrm{where} \phantom{x} p \phantom{x} \textrm{is prime.}
\]
I mentioned that such primes are relevant to the problem of finding perfect numbers. So how about we flesh that out?
Let's define a function that takes in positive integers and records the sum of their divisors. We shall denote this function with the Greek letter sigma, so that
\[
\sigma(n)=\sum_{d|n} d = \phantom{x} \textrm{the sum of the divisors of} \phantom{x} n.
\]
Here are some sample values:
\[
\sigma(3)=1+3=4…
jrosenhouse | April 25, 2011Like all moderately curious people, I'm sure you've often wondered whether it's possible for
\[
N=a^k-1
\]
to be a prime number, where a and k are positive integers. Well, I'm here to answer that for you!
To avoid trivial cases, we shall assume that k is at least two. Of course, I'm sure we all remember the basic factorization formula:
\[
a^2-1=(a+1)(a-1),
\]
which can only be prime if a=2. A prime number can only be written as a product of two others if one of those numbers is one, you see.
Let's grind out a few cases to see what happens when the exponent is three or four:
\[
3^3-1…
jrosenhouse | April 21, 2011As soon as I am done teaching this afternoon I will hop into the Jason-mobile and sally forth to scenic Roanoke, Virginia. I will be giving my Monty Hall talk at Roanoke College this evening at 7:00. If anyone reading this is familiar with the campus, I will be speaking in Massengill Auditorium. See you there!
jdupuis | April 10, 2011I have a whole pile of science-y book reviews on two of my older blogs, here and here. Both of those blogs have now been largely superseded by or merged into this one. So I'm going to be slowly moving the relevant reviews over here. I'll mostly be doing the posts one or two per weekend and I'll occasionally be merging two or more shorter reviews into one post here.
This one, of Super Crunchers: Why Thinking-By-Numbers is the New Way To Be Smart, is from April 12, 2008.
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You know how I'm always complaining about business-y buzz/hype books & articles? How they're 1/3 repetition,…
mstorksdieck | April 1, 2011Within certain education and policy circles the acronym STEM (i.e., science, technology, engineering, math) has become a common term, used frequently to be inclusive when referring to a broad area of scholarship and enterprise we deem particularly connected, i.e., those listed four subjects. How, or even whether the acronym is understood and fashionable outside these education "insider" groups is not well know. What is known, though, is that the acronym and associated term is not well defined even within groups that make heavy use of it.
When we say STEM, do we simply mean any of the four…
jrosenhouse | March 30, 2011As a professional mathematician I get to visit a lot of scenic tourist destinations. Like Slippery Rock, Pennsylvania! I'll be giving a colloquium talk to the math dept. at Slippery Rock University tomorrow afternoon. The topic? The Monty Hall problem. Surprise!
See ya when I get back.
jrosenhouse | March 30, 2011My number theory course has recently moved on to things that are a bit more technical and esoteric than our earlier fare, so I haven't felt they would make good blog fodder. If you need a quick math fix (and who doesn't?), you can have a look at this guest post I wrote for the Oxford University Press blog. It contains a few musings about pi, inspired by a recent satirical post over at HuffPo. Enjoy!
mikethemadbiologist | March 15, 2011By way of Observational Epidemiology, we find an interesting NY Times article by Michael Winerip describing a seventh grade teacher's experience with value added testing in New York City. I'll get to value added testing in a bit, but the story also highlights why we need more reporters who have backgrounds in math and science. Winerip:
On the surface the report seems straightforward. Ms. Isaacson's students had a prior proficiency score of 3.57. Her students were predicted to get a 3.69 -- based on the scores of comparable students around the city. Her students actually scored 3.63. So Ms.…
jrosenhouse | February 28, 2011The big number theory course has moved on to modular arithmetic, which means we have been discussing Fermat's Little Theorem. Personally I've always thought that name is just adorable. As it happens, I already did a post on this topic. But since that post is close to a year old, and since I did not have my funky TeX renderer installed then, how about I repost it. Enjoy!
Fermat's Little Theorem is a classic result from elementary number theory, first stated by Fermat but first proved by Euler. It can be stated in a number of different ways, but here is the version most useful for what…
jrosenhouse | February 14, 2011The big number theory class has moved on to prime factorizations and the fundamental theorem of arithmetic. As it happens, though, I've already done a post on that subject.
Looking back at what I wrote then I see that I left out one important detail. I asserted without proof (though I did provide a link) that if a prime divides the product of two other numbers then it had to divide one of the two factors already. The proof of that result requires something that we estblished last week. So let's have a look.
Last week we mentioned that it is a consequence of the Euclidean algorithm that if…