## New WordPress email template

I recently received a notification from WordPress that they had updated the template used for sending emails out to subscribers to WordPress blogs.

I am curious to see how a bit of maths looks like in the new email template. So I should write something. (I’m subscribed to my own blog so I’ll see what the email looks like!)

I have a draft somewhere of something I was going to write about characteristic functions, counting and the inclusion/exclusion principle. Ah I see I already published Part 1 at

Another observation which can be helpful is the following. Suppose that $s$ and $t$ are elements of $\{0,1\}$. Then $s-t \in \{-1,0,1\}$, and this will be even even if and only if it is zero, i.e., if and only if $s=t$. In terms of modular arithmetic, working modulo $2$, for $s$ and $t$ in $\{0,1\}$, we have

$s=t \Leftrightarrow s \equiv t~~(\mathrm{mod}~2)\,.$

Here the forward implication is trivial, but the backward implication uses the restriction on $s$ and $t$.

Since characteristic functions only take the values $0$ or $1$, we have the following extension of the above. Let $A$ and $B$ be subsets of $X$. Then (I must find a better way to write multiline LaTeX with alignment in WordPress! Perhaps they allow this by now and I just haven’t noticed?)

$A = B \Leftrightarrow \chi_A = \chi_B$

$\Leftrightarrow$ for all $x \in X$, we have $\chi_A(x)=\chi_B(x)$

$\Leftrightarrow$ for all $x \in X$, we have $\chi_A(x)\equiv\chi_B(x)~~(\mathrm{mod}~2)\,.$

Here it is convenient to bring in some new terminology to save some writing.

Let $f$ and $g$ be integer-valued functions defined on $X$. We say that $f$ is congruent to $g$ modulo $2$, and write $f \equiv g ~~(\mathrm{mod}~2)$ if (and only if), for all $x \in X$, we have

$f(x)\equiv g(x) ~~(\mathrm{mod}~2)\,.$

In other words, we require $f(x)-g(x)$ to be even for all $x \in X$.

To be continued!

## Measure Theory playlists on MediaSpace and YouTube

I have now finished correcting the captions of my 15 Measure Theory episodes. This material was one of the chapters from the Level 4 module Further Topics in Analysis, as I taught it back in 2011-12.

My Measure Theory page on this blog now links to the MediaSpace versions of these videos, which should also be viewable in China (in case you have contacts in China who want to see these!). They have also been released on the main University of Nottingham YouTube channel.

## Regularity conditions for Banach function algebras

On this blog I usually talk about mathematics topics suitable for maths undergraduates, maths A-level students or (sometimes) GCSE maths students. I have been involved in maths outreach for far younger school students. And at the other extreme I do give research seminars for specialists in my area of Functional Analysis (Commutative Banach Algebras).

## Measure Theory captions finally complete

Well, it took a lot of time, and a lot of laughter, but I have finally finished correcting the auto-generated captions for my Measure Theory videos: 15 episodes on MediaSpace, with 14 so far published with corrected captions on YouTube. The last episode will soon be published on YouTube with corrected captions.

## Maths videos on MediaSpace

Because it can be hard for people in China to access YouTube, many University of Nottingham videos are also available through MediaSpace. (See https://mediaspace.nottingham.ac.uk/)

For example, most of my old Measure Theory videos are available there (12 out of 15 so far. I’m still correcting captions for the last few!)

## Captions for Measure Theory and Beyond Infinity

After a long break, I have returned to correcting the subtitles/captions for some of my older videos!

I think that there is a convention that captions are for the same language, and subtitles are for a different language. (If that’s right then the people who published lots of my videos in China have provided English captions and Chinese subtitles.)

I started with making minor corrections to the captions for the Virtually Nottingham edition of Beyond Infinity (the most recent edition of my talk about Hilbert’s Hotel, with quite a long Q&A session afterwards). It looks like my corrections have been incorporated already, but I don’t yet know whether this was automatic or else very efficient work by a colleague!

Now I’m back to my old Measure Theory videos. You may remember that (perhaps due to a combination of me speaking too fast, and questionable audio quality) there were a lot of very funny errors in the automatically generated captions. I know that I went over the top with that list! But let me reveal some of the answers. (I’ll add some more later when I can remember the answers myself!)

Of course, I keep finding more, but I’ll resist posting most of them (unless more are requested).

Still, I just can’t resist telling you that “the Bogan’s robots” are supposed to be de Morgan’s laws!

I’ll let you know when the new versions of the Measure Theory videos with captions are released on YouTube.

## University of Nottingham Maths Taster Sessions YouTube Playlist

Thanks to my colleague Jamie Walton, we now have a UoN Maths Taster Sessions PlayList on YouTube at https://tinyurl.com/UoNMathsTaster-YT

The Playlist description currently says:

We have been running a series of Taster Lectures and Popular Maths Talks. You can find more information, including how to join us live for future taster sessions, at this link: https://tinyurl.com/uonmathstaster

– Taster Lectures give you a taste of what a maths lecture at the University of Nottingham is really like, often based on content that we teach our first-year students. These are mainly suitable for Year 12/13 A level maths students.

– Popular Maths Talks give you the opportunity to hear one of our university teachers talk about an exciting mathematical topic, and how it links to the maths we teach at Nottingham. This could range from how to use mathematics in the fight against Covid-19, to climate change predictions. These are still most suitable for Year 12/13 A level maths students. Younger pupils such as Year 10-11 GCSE maths students who are considering taking maths A level will also find them of interest.

## Xournal++ revisited

I just upgraded to xournal++ version 1.1.1 (stable version) to see if the problems I had before had improved. I’m pleased to say that they have! The inking now looks good, and my touch gestures for pan and zoom work properly on my Windows Surface. I’ll practice a bit more, but I think that I will probably switch over from DrawBoard PDF to xournal++ as my regular annotator.

It is true that you don’t save directly as a PDF, but “Export PDF” from the file menu works well for me for when I need a PDF file to upload/send.

I’ll continue experimenting and practising! But there are loads of great features.

## Introduction to Modular Arithmetic, Part 3b

See all posts in this series

In this post we’ll look at some applications of the result from last time that we gave the non-standard name Modular Arithmetic Consistency Theorem, or MACT for short. In particular, we will finally have a proper look at powers of integers.

Theorem (MACT)

Let $k$ be a positive integer, and let $a_1$, $b_1$, $a_2$ and $b_2$ be integers. Suppose that $a_1\equiv a_2~~(\mathrm{mod}~k)$ and $b_1\equiv b_2~~(\mathrm{mod}~k)$. Then

$a_1+b_1 \equiv a_2+b_2 ~~(\mathrm{mod}~k)~~~ \textrm{and}~~~ a_1b_1 \equiv a_2b_2 ~~(\mathrm{mod}~k)\,.$