Credits: Anne Margo & Australian Broadcasting Commission

Somewhere over the rainbow (way up high).

Credits: BEVK12 & Australian Broadcasting Commission

This guy’s situation reminds of the Dusty Springfield song; “ Wishin’ and hopin’ and thinkin’ and prayin’ Plannin’ and dreamin’  …”

I love the book “Liquid Millionaire”. I hope he makes it.

Credits: Jon Hanrahan & Australian Broadcasting Commission

I know, the programming language Python is named for Monty Python, but I think the picture is funny. She’s certainly got control of that snake!

Jim Tanton (@jamestanton) posed the problem of 6 positive integers such that the product of any two of them is a multiple of their sum (preferably 6 integers with no common factors).

Inspired by Jim’s question I asked myself what happens for 3 integers. This led to the following simple dynamical system:

Given a triple of integers (not necessarily positive) replace this triple by

unless , in which case leave as .

This gives us a dynamical system on the lattice of integer triples.

The orbit of a triple consists of all triples obtainable from   by the above scheme.

Playing around

For example, the orbit of is

Whereas, the orbit of leads, in 22 steps, to the 2-cycle

The starting point leads to a 6-cycle in just 1 step.

The orbit of has length 57 , ending in a 6-cycle.

Questions

1. For which triples  does the orbit contain ?
2. For which triples does the orbit lead to a fixed point other than ? What are the possible fixed points?
3. Ditto, what are the possible period 2 points – ones that flip by replacement- and which starting triples lead to them?
4. What periods are possible for periodic points?

Have fun!

In the video below, Aziz Belhassane Sensei is demonstrating Aikido at the Sugano Shihan Memorial Summer School 2011.

Whether you know Aikido or not, the thing you will notice, especially if I point it out to you, is that Aikido is not a spectator activity. You cannot learn Aikido simply by watching someone else. Watching is critical, paying attention is critical, but doing is the essence.

Currently I am teaching a small class – 11 students – what is known in the U.S. as Advanced Calculus. This is their second semester of Advanced Calculus, and I am conceiving it as an introduction to real analysis. This past week we have been looking at the Cantor function (also known as the Devil’s Staircase).

The Cantor function can be described as the limit of a sequence of piecewise linear functions. The first 3 are shown below:

The piecewise linear functions can be defined on the interval recursively as follows:

So we have a potential pictorial and recursive understanding of the sequence of approximations to the Cantor function.

These do not make it immediately obvious how to calculate the value of the Cantor function at a given point in the interval , particularly at points of the Cantor set, such as .

A simple recipe remedies this:

Write base 3 (with no all 2′s from some point on ). If one of the replace all digits after the first such occurrence by 0. Then convert all occurrences of 2 to 1 and read the new number base 2.

For example, base 3, so, according to the above recipe,  the value of the Cantor function at is base 2, which is .

The issue for the students was to reconcile the pictorial and recursive approximations to the Cantor function with this recipe.

Sadly, as was the case for most of the semester, they sat in silence waiting  for me to do something, even though the course is seminar style in which they present their efforts at problem solving. So I gave them all marker pens, got them up to the board, and got them working and talking. Eventually most of them started thinking.

In my view, mathematics is best done on your feet, actively moving.

What do you think?

The Geogebra folks continue to work at developing the capabilities of this lovely open source software. In February this year they added a spreadsheet capability to the software

I have a bunch of fairly naive questions about the use of Geogebra for statistics. Probably the main questions are:

• Who is using it to teach statistics, and at what levels (elementary school, middle school, high school, college, graduate school ?)
• Who is using it for data analysis, and in what contexts?
• What size data sets can Geogebra handle?

If anyone has even partial answers to these questions I’d be glad to hear about them. I’d also like to hear any and all stories of use statistical use of Geogebra, and I’m happy to share those stories here.

I’m aware that Geogebra has a bunch of statistics commands and functions: I’m interested in how people are using Geogebra for statistics.

Thanks for your time,

Gary Davis

This is a follow-up to the recent post Mathematics majors: are most of you wasting time and money?

The following post was made on WrongPlanet during the height of the Great Recession, so it’s perhaps a little understandable that the writer was finding it hard to get a decent job, or any job at all.

Posted: Sun Jan 18, 2009 2:33 am Post subject: Graduated with a Math degree… feel like a moron.

“So as the title says, I recently graduated with a degree in Mathematics. Joy of joys… it’s supposed to be a proud moment in my life. But now my situation is sinking in and I can’t find anything to do with my new-fangled degree.

Nothing in the job listings seems applicable to my skills, or they require some technical jargon laden skill set. Or there are accounting or business or financial jobs where I would have to wear a suit and all the office politics. I don’t even own a suit or understand these work environments.

It seems like I’ll have to go get a factory job, grocery store or Wal-Mart type job. But that type of employment seems counter to all the expectations and self esteem and what have you that has been building up in me over the last 26 years.

I don’t even know what I’m writing this for. I have two parts to me… parts that are too smart… and parts that are too dumb.

I’m glad I had a good time studying math… I might as well have spent those 6 years getting drunk and doing drugs….Because now I’m paying the price for studying what I like and not heeding the advice of those who said that there was really no jobs for those who study math exclusively.”

What do you think mathematics majors? Is this your experience?

Evolution of the 2 glyph:

The 4! = 24 distinct fractions whose base 4 representation consists of permutations of the digits (0,1,2,3) repeated:

These are all fractions of the form , namely those fractions  with numerators 9, 10, 13, 15, 18, 19, 25, 26, 33, 36 ,38, 40, 45, 47 ,49 ,52, 59 ,60 66, 67, 70, 72, 75 and 76.

Below is a post on College Confidential, posted by mathematik on May 24, 2010:

“Math majors beware: most of you are wasting time and money

Here’s a story for all the math majors and future math majors reading this:

I graduated with a bachelor’s degree in math 6 years ago. I graduated with honors and a good GPA from a good college. My current job prospects are horrible. I am stuck at a dead end programming job where I earn less than $40,000 per year even though I live close to a major US city and have over 5 years of work experience. I make less money than my “dumb” friends from high school currently make.

Perhaps I’m to blame for thinking that being good at math meant I was intelligent, but I feel morally obliged to warn people reading this who might be on the same boat. My honest suggestion to you is to major in something else. Unless you want to be a math teacher or an actuary, I implore you not to listen to anyone who tells you that you can do a million things with a math degree and that you should major in anything you want and the money will follow. Those are cliches and empty promises that don’t mean anything in the real world.

You could argue that you could major in math and possibly end up with a programming job, since I did it, but don’t expect to get hired to work as a software engineer unless you have additional qualifications, expect to work doing quality assurance or fixing code. It doesn’t matter how smart you think your degree in math makes you, you can’t compete with the millions of computer science majors in this country when it comes to programming jobs.

If you like finance, accounting, science, engineering, or some other subject, why not major in that subject? Why waste your time and money learning math formulas that will be of no use to you in the real world? If you major in math, and end up working as an accountant, don’t you think you would be a more competitive candidate for accounting jobs if you had majored in accounting?

That’s my advice to you guys. Do whatever you want with your lives but if you end up like me don’t say that you were not warned.”

What do you think BA/BS mathematics graduates?

Does your experience match that of mathematik?

Or have you had more career success with your mathematics degree?