Frag doch mal die Maus!

As part of our attempt to bring up our children to be bilingual, we all mostly watched German TV when the children were young. One particularly good programme was (and is) the entertaining and informative “Die Sendung mit der Maus”. I learned something new every week from that programme!

This evening we watched some of a spin-off evening quiz programme Frag doch mal die Maus! (Currently available at http://www.wdrmaus.de/maus_wall/frag_doch_mal_die_maus.php5?detail=080420162015The questions are quite interesting and fun.

At one point the teams had to estimate the number of blades of grass (main stem only) on a football field. I won’t reveal the “official” answer they suggested, but I will say that the two teams’ estimates differed by a factor just over 40. But how should you judge the winner here? In this case, the lower answer was “closer” numerically, but not of the correct order of magnitude, while the other answer was of the correct order of magnitude, but further from the official answer numerically. I thought that answer was better, but I think the other answer took the points.

(Would it be better to take logs?)

 

Challenging General Relativity

My colleague Thomas Sotiriou has sent me the following information about recent public engagement outreach events he has been involved in, with videos available.

We had two outreach events in the last few months. One was about gravitational waves and the other was a series of four public talks celebrating the centennial of General Relativity. All talks for both events have been recorded and can be found here (together with more info about the events)

http://thomassotiriou.wix.com/challenginggr#!outreach/c10aq

Swiss cheeses

My readers may know that I do a lot of research on Swiss cheeses (though being mathematical, they tend to have infinitely many holes) [Note added: the Swiss cheeses, not the readers!].

 

CheeseColour

My latest joint paper on Swiss cheeses with my research students Sam Morley and Hongfei Yang, Abstract Swiss cheese space and classicalisation of Swiss cheeses , has just been published in Elsevier’s Journal of Mathematical Analysis and Applications. Thanks to funding from the EPSRC, this article has been made Open Access, so anyone can access the final published version free of charge at

http://dx.doi.org/10.1016/j.jmaa.2016.02.004

Of course much of the material is beyond the level of the typical undergraduate course. Nevertheless, students in 3rd/4th year might get something from looking at this. Abstract Swiss cheese space itself is really just a product of a sequence of standard spaces, but with elements interpreted as sequences of centres and radii of “abstract” discs. The two basic elementary geometric lemmas could probably be taught at GCSE!

 

Selected links

I have compiled a somewhat biased selection of links which I issue at the maths outreach events I run at the University of Nottingham.

Here is what I provide at the moment!
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Some useful and interesting links, selected by

Dr Joel Feinstein, Outreach Officer,

School of Mathematical Sciences,

The University of Nottingham

You may also find it useful to look up some mathematical topics on Wikipedia!

 

Working with definitions

When I set a question asking students to prove things, the most common question I get is “How do I start?” Further investigation often reveals that the students don’t remember the definitions of the terms that appear in the question, and have not developed fluency in one relatively routine aspect of doing proofs: make sure that you know what the information you have been given means.

For those students on my first-year, first-semester module on pure maths, mathematical reasoning using definitions, proofs and examples is completely different from what they have seen at A level. My emphasis on the importance of precise definitions is seen by some as rather dry. But I am trying hard to help students to practice working with definitions, proofs and examples as much as possible for themselves.

In my first-year workshop this week, I had a question asking students to prove that if you compose two injections you get an injection, and if you compose two surjections you get a surjection. It took me a few minutes to realise that I was going to have to write the definitions of injection and surjection up at the front, because that was where most people were stuck. I’ll probably include a reminder on the worksheet next time!

The “How do I start?” issue (relating to working with definitions) is not just a first-year phenomenon: it persists through second and third year, and not just in workshops, but also in coursework. I think that I’m going to have to find some new, more exciting, ways to get this idea across, and to try to help students to get past this initial obstacle so that they can spend their time thinking about the interesting bits instead!

Modular arithmetic, number theory and encryption

Here is a message I just sent to my first-year students at Nottingham ….

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Hi everyone,

You have now seen a bit of modular arithmetic, which you can also think of as introductory number theory. Prime numbers and modular arithmetic are the key to the “public key” encryption systems used on the internet whenever you visit a secure website (https). You can learn more about all this in later modules.

Today’s classes included some tricky concepts. Many of you will still be wondering about “squares modulo 7”.

Perhaps modulo 10 is a bit easier to think about here? Remember that, for non-negative integers, working modulo 10 is like ignoring everything except the last digit, so all non-negative integers end up being treated as being between 0 and 9, and you have multiplication tables including 9×6=4, 2×5=0, 8×7=6, etc. (mod 10).

So what are “squares modulo 10”? Well, what are the possible last digits of squares of integers?

First note that squares of negative numbers are just the same as squares of positive integers, so there is no need to check those.

Checking the last digits of the squares of 0, 1, 2, 3, … we see the pattern

0, 1, 4, 9,6,5,6,9,4,1,0,1, 4, 9,6,5,6,9,4,1,0,…

and this repeats forever. Note that 4^2 and 6^2 end in 6, 3^2 and 7^2 end in 9, etc. This is not a coincidence, because 6 is congruent to -4 modulo 10, so 6^2 is congruent to (-4)^2=4^2 (mod 10).

The repeating of the pattern with period 10 is also no surprise, because (n+10)^2 is congruent to n^2 (mod 10).

Conclusion: no matter which integer you square, the final digit of the square will be one of 0, 1, 4, 5, 6 or 9. You can’t get anything else.

This is because we are using the decimal system. If you use octal (base 8) instead, the same thing happens when you work modulo 8: for non-negative integers, just look at the last base 8 digit. Modulo 8 you get 3×3=1, 2×4=0, etc.

I hope this helps a bit!

Best wishes,

Dr Feinstein

Banach Algebras 2015

I’m currently at the 23rd international conference on Banach Algebras. This year we are at the Fields Institute in Toronto.

They are recording videos of all of our talks: see  http://www.fields.utoronto.ca/video-archive/event/394/2015

My talk isn’t there yet (at the time of writing), but the talk of Sam Morley (one of my PhD students) is ready.

There have been lots of excellent talks, as you would expect from the list of speakers.

Now I just have to dodge the thunderstorms this evening …