My latest post on my Blogger blog is about when you should use the Ratio Test (for sequences and/or series) and, perhaps more importantly, when you should definitely NOT use the Ratio Test. I describe this briefly as a rule of thumb for when to use the Ratio Test

For more of these summaries, you can search this blog for “Latest posts on my Blogger blog”, or directly see https://explainingmaths.wordpress.com/?s=%22Latest+posts+on+my+Blogger+blog%22

Here are some links to my most recent posts on my Blogger blog, where I am able to use MathJax.

I’ll periodically post updates on this blog, with links and brief descriptions.

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I’ve recently finished teaching 5 weeks of Analysis as part of our new first-year Core Mathematics module. I’ve made some additional resources available to them on a Moodle page. I expect to add further resources in due course! But here is the current version of that page.

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For more of these summaries, you can search this blog for “Latest posts on my Blogger blog”, or directly see https://explainingmaths.wordpress.com/?s=%22Latest+posts+on+my+Blogger+blog%22

Here are some links to my most recent posts on my Blogger blog, where I am able to use MathJax.

I’ll periodically post updates on this blog, with links and brief descriptions.

Continue reading →

For more of these summaries, you can search this blog for “Latest posts on my Blogger blog”, or directly see https://explainingmaths.wordpress.com/?s=%22Latest+posts+on+my+Blogger+blog%22

Here are some links to my most recent posts on my Blogger blog, where I am able to use MathJax.

I’ll periodically post updates on this blog, with links and brief descriptions.

Continue reading →

One of our first-year students asked (on Piazza) whether converse and negation were the same thing. One of my colleagues explained the differences in terms of propositional logic. I added some comments afterward to see if some specific examples might help. I don’t know whether this helped or not! See below for what I said.

This post is available from my Blogger blog at
https://explaining-maths.blogspot.com/2023/10/converses-and-negations.html
Alternatively, a PDF of that post is available for viewing or downloading below.

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My recent series of posts about Fatou’s Lemma for sums was aimed primarily at mathematics students at 3rd-year undergraduate level or above. So I think I should write a post more suitable for first-year undergraduate students. So I’m going to have another look at the topic of and for bounded sequences of real numbers (though most of what I say can be generalised to sequences, or even nets, of extended real numbers).

This post is available from my Blogger blog at
https://explaining-maths.blogspot.com/2023/08/revisiting-liminf-and-limsup.html
Alternatively, a PDF of that post is available for viewing or downloading below.

See also my earlier post
An application of absorption to teaching lim inf and lim sup (sequences)

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

See https://wordpress.com/blog/2022/05/19/your-website-looks-great-so-should-your-emails/

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

Sets and their characteristic functions, Part 1

But I hadn’t had time to add the follow-up yet. So here is a sneak preview!

Another observation which can be helpful is the following. Suppose that and are elements of . Then , and this will be even even if and only if it is zero, i.e., if and only if . In terms of modular arithmetic, working modulo , for and in , we have

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

Since characteristic functions only take the values or , we have the following extension of the above. Let and be subsets of . 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?)

for all , we have

for all , we have

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

Let and be integer-valued functions defined on $X$. We say that is congruent to modulo , and write if (and only if), for all , we have

In other words, we require to be even for all .

To be continued!

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!)

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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!)

Autogenerated captions	What I think I really said (or tried to say)
… et cerebral llamas	… extended real numbers
… topology or eggs	… topology on X
… bomb umbrella sets	… non-Borel sets

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.

I have mentioned our University of Nottingham Mathematics YouTube channel at https://tinyurl.com/uonmathsyt (have you subscribed to our YouTube channel yet?) and our Taster Sessions PlayList at https://tinyurl.com/UoNMathsTaster-YT

But the University of Nottingham also has its own channel at https://www.youtube.com/c/uniofnottingham

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If you saw my last post here you will know that I recently took advantage of editing our UoN maths newsletter to write an article about Alice Roth and Swiss cheeses.

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