This week’s entry on what-if.xkcd.com contained a pointer to the blog of Derek Lowe: in particular, to his hilarious series of posts on insanely dangerous chemicals aptly summarized by the tag “Things I Won’t Work With”. To all my science geek friends, I strongly recommend
reading the series, and
not attempting to do so while eating, drinking, or sitting on a chair.
In the last few days I’ve managed to pass another milestone in terms of post-operative recovery: being able to flex my knee enough (and without significant pain) to sit Zazen in Burmese posture. This little achievement means a lot to me, since being able to sit in a stable position without stressing one side more or less than the other is a big help in straightening everything out: having one dodgy knee leads to all sorts of imbalances in posture, legs, the spine, &c. Day by day, progress feels frustratingly slow, but I have to remember that I was nowhere close to sitting like this a week or two ago. Again — perspective!
As I’m having lunch in a nice little café, one of the staff is busy showing a new boy the ropes. I can overhear him explaining that, when it comes to keeping the front of the house clean and tidy, it’s all about the customer’s perspective. Together, they wipe this, straighten that, and then — as he says to his colleague — “Just step back and see it from the customer’s perspective.” It’s a good philosophy, I think. As Robert Burns wrote in his poem To a Louse, On Seeing One on a Lady’s Bonnet at Church,
O wad some Pow’r the giftie gie us
To see oursels as ithers see us!
It wad frae mony a blunder free us,
An’ foolish notion:
What airs in dress an’ gait wad lea’e us,
An’ ev’n devotion!
The trouble with the senior waiter’s fine philosophy is that, from this customer’s perspective, the most notable thing about him is that his neat appearance (black shirt, black trousers, black apron) is totally undercut by the fact that his trousers are hanging down below his hips, exposing his light grey underpants to the world.
I just attended a very fun talk by Saul Schleimer on the 3D printing of various mathematical shapes. Since last year, he and his collaborator Henry Segerman have been producing a fascinating array of mathematical shapes such as the triple gear, various cell structures, knots and surfaces. Segerman has a selection of videos on his YouTube page. These sculptures provide a wonderful visualisation of many intricate mathematical structures, as well as simple aesthetic pleasure and even some puzzles to rival Rubik’s cube!
One of the theorems that I make frequent use of in my uncertainty quantification (UQ) research concerns probabilities measures on Radon spaces. Without going into details, the UQ methods that I like to work with will work fine if the spaces where your uncertain parameters / functions / models / other gubbins live are Radon spaces, and might fail to work otherwise; therefore, it’s important to know what a Radon space is, and if it’s a serious restriction. (It’s also very useful to know some pithy examples use in response to questions about Radon spaces in talks and poster presentations!)
So… the definition. Consider a topological space (X, T) and a probability measure μ: ℬ(T) → [0, 1] defined on the Borel σ-algebra ℬ(T) (i.e. the smallest σ-algebra on X that contains all the open sets, i.e. those sets that are listed in the topology T). The measure μ is said to be inner regular if, for every ℬ(T)-measurable set E ⊆ X,
This is often informally read as saying that the measure of an arbitrary measurable set can be approximated from within by compact sets. The space (X, T) is called a pseudo-Radon space (my terminology) if every probability measure μ on ℬ(T) is inner regular, and if the space is also separable and metrizable then it is called a Radon space (more standard in the literature, e.g. the book of Ambrosio, Gigli & Savaré on gradient flows in metric spaces).
So, what spaces are (pseudo-)Radon spaces? It turns out that most of the “nice” spaces that one might want to consider are Radon:
any compact subset of n-dimensional Euclidean space ℝn is Radon,
indeed, Euclidean space itself is Radon,
as is any Polish space (i.e. a separable and completely metrizable space),
as indeed is any Suslin space (i.e. a continuous Hausdorff image of a Polish space).
This all seems to suggest that non-Radon spaces must be very weird beasts indeed, perhaps spaces that are topologically very large, so much so that they cannot be the image of a separable space. However, there are, in fact, “small” examples of non-Radon spaces. Since just one counterexample will suffice, it’s enough to find a single example of a non-inner-regular measure on space to show that it is not (pseudo-)Radon.
Art is a way of saying what it means to be alive, and the most salient feature of existence is the unthinkable odds against it. For every way that there is of being here, there are an infinity of ways of not being here. Historical accident snuffs out whole universes with every clock tick. Statistics declare us ridiculous. Thermodynamics prohibits us. Life, by any reasonable measure, is impossible, and my life — this, here, now — infinitely more so. Art is a way of saying, in the face of all that impossibility, just how worth celebrating it is to be able to say anything at all.”
Over the last week or so, as my recovery from last month’s knee surgery has continued, I’ve been making use of one of my “dormant” martial arts to aid my healing and re-conditioning process: taijiquan. The slow, controlled movements are ideal for my still-weak knee: nothing too dramatic or demanding, but it’s easy to turn up the difficulty by small degrees simply by doing the movements slightly lower to the ground — and, by goodness, does that work and strengthen the legs nicely!
I studied Taijiquan for two years as an undergraduate student. At the time, I was practising Taijiquan more than Aikidō, and hadn’t even encountered Iaidō yet; things changed after the fellow university student who led the Taijiquan classes graduated and left the university. Even so, Taijiquan left its imprint upon me. Also, with increasing experience in other martial arts, I feel the martial and conditioning content of Taijiquan more and more strongly. Right now, it’s the conditioning aspect that I’m making most use of. However, it’s also ever clearer to me that Taijiquan is an excellent example of things not being what they appear to be on the surface: as soft and fluffy as it may appear to be, Taijiquan is in fact a tremendously strong and strengthening martial practice, something that I have long known but now increasingly feel.