Latest posts on my Blogger blog 6/4/24

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.

My latest set of posts is a series of 6 posts (intended for mathematics undergraduates in their 3rd or 4th year) on the Hahn-Banach extension theorem for continuous linear functionals on normed spaces.

Date	Title and link	Description
7/4/24	An introduction to the Hahn-Banach extension theorem: Part VI	Proof of the real Hahn-Banach theorem using transfinite induction (some details left to the reader).
6/4/24	An introduction to the Hahn-Banach extension theorem: Part V	Proof of the real Hahn-Banach theorem using Zorn’s Lemma
2/4/24	An introduction to the Hahn-Banach extension theorem: Part IV	How every complex normed space is also a real normed space, and how the complex and real dual spaces are related. Deduction of the complex Hahn-Banach theorem from the real Hahn-Banach theorem.
1/4/24	An introduction to the Hahn-Banach extension theorem: Part III	The Hahn-Banach Extension Theorem for linear functionals on separable, real normed spaces, proved using the key lemma and (at the end) extension to the closure.
31/3/24	An introduction to the Hahn-Banach extension theorem: Part II	– A look at the operator norm for bounded linear functionals on real normed spaces. – The key lemma used to extend bounded linear functionals on real normed spaces by one dimension without increasing the operator norm.
31/3/24	An introduction to the Hahn-Banach extension theorem: Part I	The statement of the Hahn-Banach extension theorem for bounded linear functionals on normed spaces, along with some initial comments about how we can approach the proof.

Latest posts on my Blogger blog 16/3/24

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.

Date	Title and link	Description
16/3/24	An introduction to the Weierstrass M-test: Part IV	Differentiation of power series term by term, based on the Weierstrass M-test.
14/3/24	An introduction to the Weierstrass M-test: Part III	A look at differentiation of series of functions term by term, based on the Weierstrass M-test.
11/3/24	A Cauchy-L’Hôpital-Taylor hybrid theorem	A two-function version of Taylor’s Theorem, from which you can deduce the usual version of Taylor’s Theorem with (Lagrange) remainder.
18/2/24	An introduction to the Weierstrass M-test: Part II	Proof of the Weierstrass M-test for series of real-valued functions defined on an interval: convergence and continuity of the resulting function under suitable conditions.
17/2/24	An introduction to the Weierstrass M-test: Part I	A preliminary look at series of real-valued functions defined on an interval, hopefully accessible to first-year mathematics undergraduate students.

Optional additional resources for first-year analysis

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.

Further Topics in Analysis optional additional resources

My Level 4 analysis module has been through various codes and titles. Currently it is MATH4047 Further Topics in Analysis (FTA for short).

I have made some optional additional resources available to them via a Moodle page.

Here is a copy of that page!

Latest posts on my Blogger blog 11/3/24

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.

Date	Title and link	Description
11/3/24	A Cauchy-L’Hôpital-Taylor hybrid theorem	A two-function version of Taylor’s Theorem, from which you can deduce the usual version of Taylor’s Theorem with (Lagrange) remainder.
18/2/24	An introduction to the Weierstrass M-test: Part II	Proof of the Weierstrass M-test for series of real-valued functions defined on an interval: convergence and continuity of the resulting function under suitable conditions.
17/2/24	An introduction to the Weierstrass M-test: Part I	A preliminary look at series of real-valued functions defined on an interval, hopefully accessible to first-year mathematics undergraduate students.

8×8 Othello “solved”

This is amazing! I never thought I’d see this without some major breakthrough in computational power. It looks like, as long suspected, 8×8 Othello is a draw with perfect play.

https://arxiv.org/abs/2310.19387#:~:text=The%20challenge%20of%20solving%20Othello,players%20lead%20to%20a%20draw

See https://www.maths.nottingham.ac.uk/plp/pmzjff/Othello/6x6sol.html and https://www.maths.nottingham.ac.uk/plp/pmzjff/Othello/Amenor.html for what I established back in 1993 with my own (manually tuned, but somewhat amateurish) software on what was then our fastest departmental computer.

White to play and win 20-16!

Latest posts on my Blogger blog 18/2/24

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.

Date	Title and link	Description
25/12/23	The Christmas Equation	Taking a short break from total boundedness and Cauchy sequences, this is a repost of some funny maths I saw on QI a few years ago
25/12/23	Total boundedness for countable metric spaces	Some musings on the special case of countably infinite metric spaces. Not particularly conclusive! But there are various reformulations of the results above, and a note that imposing a well-order enables us to do without the usual sequence of choices. Also some discussion of uniformly separated subsets.
26/12/23	Conclusions on Totally Bounded metric spaces and Cauchy sequences	A summary of all the facts from the posts above, relating total boundness, uniformly separated sequences and subsets, and Cauchy subsequences
26/12/23	Other approaches to connected components (from this year’s MATH4085 Metric and Topological Spaces)	An alternative approach to connected components, based on considering unions of just two connected sets with non-empty intersection, and then defining a suitable equivalence relation. (See also the earlier post on this.)
17/2/24	An introduction to the Weierstrass M-test: Part I	A preliminary look at series of real-valued functions defined on an interval, hopefully accessible to first-year mathematics undergraduate students.
18/2/24	An introduction to the Weierstrass M-test: Part II	Proof of the Weierstrass M-test for series of real-valued functions defined on an interval: convergence and continuity of the resulting function under suitable conditions.

Measure and Integration 2006-7-8 update

Thanks to the local team at the University of Nottingham for helping me to track down my missing Measure and Integration mp3 files. I’m pleased to say that they were all safely backed up. I am now hosting the mp3 files on this blog.

Yesterday and today I’ve been working on the old Measure and Integration module page (from 2006-7-8) at http://tinyurl.com/UoNG13MIN, and I think that I have updated most of the links. But I might have missed a few, so please let me know if you spot any broken links!

Screencast and audio links from this blog.

Hi everyone,

An old server was recently decommissioned here at Nottingham (I was warned!)
I thought that I had safely transferred all of my screencasts and podcasts to somewhere sensible, and to some extent that is true. What I had forgotten was that this blog still has lots of links to the old server.

It’s going to take some time to sort all this out! But the screencasts (Foundations of Pure Mathematics, Mathematical Analysis, Functional Analysis, Measure Theory) are available on YouTube, and in some cases in a number of other places.

The audio files from Measure and Integration 2006-7-8 may be harder for me to track down, so I’ll have to ask for your patience on that. Hopefully they are not lost forever.

Note added 14/2/24

Thanks to the local team for helping me to track down the files, which were safely backed up!
I have (I think!) uploaded all of the Measure and Integration mp3 files to this blog, and have also set up a tinyurl to help me update the links. For example, the old url http://wirksworthii.nottingham.ac.uk/Podcasts/files/147680/MIN-1-2-08-c.mp3 (which no longer works for me) is now updated to be https://tinyurl.com/UoN-MIN-mp3/min-1-2-08-c.mp3 (which links to the mp3 file stored on this blog).

If you have any old links for my old Measure and Integration mp3 files, the main part of the URL should now be https://tinyurl.com/UoN-MIN-mp3/ followed by the file name … but also the filename is in lower case (so MIN becomes min, Lecture becomes lecture, and MP3 becomes mp3).

Nottingham Symphony Orchestra Concert 9th March 2024 (OT)

My wife Uta and I both play violin in the Nottingham Symphony Orchestra.

Our next concert, An evening of European Classics, will be on 9th March 2024, in the (Nottingham) Albert Hall.

Here is our flyer (as screenshots and, further down, as a PDF).

1739-nso-flyer-euro-classic-march-2024Download

Piazza “less than” bug?

There appears to be a bug in (or a feature of) the Piazza mobile app (on android at least) that causes posts to be truncated at any strict less than sign inside maths.

It looks fine on a desktop web browser. (It is probably html-related.) But don’t try switching to the mark-up editor (even on the desktop web browser) as it literally deletes everything from the less than sign onwards!

Typing < inside maths in the Rich text editor on the desktop browser doesn’t help, because Piazza just converts it to the strict less than sign, which then breaks the app again.

Typing < in the Plain text editor on the desktop appears to be safe both for the desktop and for the mobile app.

This might be just a temporary bug/feature, but it may be just as well to be aware!

Note added 3 Feb 2024

I think a more efficient solution is to use the LaTeX code \lt instead of the less than sign.

Interestingly the greater than sign doesn’t appear to cause trouble. But < is probably interpreted as the start if of an html tag

I have notified Piazza of the issue, so hopefully they will fix this, or issue guidance.

Latest posts on my Blogger blog 26/12/23

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.

Date	Title and link	Description
24/10/23	Converses and negations	Discussion of some specific examples to help understand the difference between converse and negation
24/10/23	Discussion of the proof that the uniform norm really is a norm	A detailed proof that the uniform really is a norm, including some comments and warnings on possible pitfalls along the way
23/12/23	Connectedness of unions	Discussion of a standard result concerning unions of connected sets, relevant to the theory of connected components of topological spaces
24/12/23	Totally bounded metric spaces	Discussion of the fact that a metric space X is totally bounded if and only if every sequence in X has a Cauchy subsequence. Also some initial discussion of uniformly separated sequences.
25/12/23	Sequences that have no Cauchy subsequences	Discussion of the fact that a sequence in a metric space has a Cauchy subsequence if and only if it has a subsequence which has no uniformly separated subsequence. Thus a sequence in a metric space has no Cauchy subsequence if and only if every subsequence has a uniformly separated subsequence.
25/12/23	The Christmas Equation	Taking a short break from total boundedness and Cauchy sequences, this is a repost of some funny maths I saw on QI a few years ago
25/12/23	Total boundedness for countable metric spaces	Some musings on the special case of countably infinite metric spaces. Not particularly conclusive! But there are various reformulations of the results above, and a note that imposing a well-order enables us to do without the usual sequence of choices. Also some discussion of uniformly separated subsets.
26/12/23	Conclusions on Totally Bounded metric spaces and Cauchy sequences	A summary of all the facts from the posts above, relating total boundness, uniformly separated sequences and subsets, and Cauchy subsequences

Discussion of the proof that the uniform norm really is a norm

In response to a request on Piazza, I gave a detailed proof that the uniform really is a norm, including some comments and warnings on possible pitfalls along the way. Here (essentially) is what I said.

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

Converses and negations

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.

Revisiting liminf and limsup

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)