Irony

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.

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Printed mathematical sculptures

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!

 

 

Saul Schleimer
Saul Schleimer holding a sculpture of half of a 600-cell. The 600-cell is the (unique!) tiling of the three-dimensional sphere S3 with regular tetrahedra, and the sculpture Saul is holding is the “northern hemisphere” of that tiling: the equator of S3 is the boundary sphere of the sculpture, and the southern hemisphere is the reflection of the sculpture in that equatorial sphere.

Radon and non-Radon spaces

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 (XT) 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,

\mu(E) = \sup \bigl\{ \mu(K) \big| K \subseteq E \mbox{ and } K \mbox{ is compact} \bigr\}.

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 (XT) 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.

Continue reading “Radon and non-Radon spaces”

Alive

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.”

Richard Powers

Recovering with the help of an old friend

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.

Three Bernstein inequalities

This post concerns three unrelated inequalities, or families of inequalities, that all go by the name of Bernstein’s inequality, after Sergei Natanovich Bernstein (Сергей Натанович Бернштейн). In no particular order, they are:

  • a family of inequalities in probability theory that bound the deviations of independent or weakly dependent random variables,
  • an inequality bounding the Lp norms of band-limited functions, and
  • an inequality bounding the derivatives of complex polynomials.

1. Deviations of Random Variables

There are quite a few Bernstein inequalities for families of independent or weakly correlated random variables X1, …, Xn. The following inequality is one of the simplest, but gives the general idea. Let X1, …, Xn be independent, but not necessarily identically distributed, real-valued random variables taking values in [−R, +R]. Then for any t > 0,

\displaystyle \mathbb{P} \left[ \sum_{j=1}^{n} X_j > t \right] \leq \exp \left( - \frac12 \frac{t^2}{\sum_{j=1}^{n} \mathbb{E} \bigl[ X_j^2 \bigr] + \frac{R t}{3}} \right).

For those readers who prefer to think in terms of the vector-valued random variable X = (X1, …, Xn), this inequality reads

\displaystyle \mathbb{P} \left[ \bigl\| \mathbf{X} \bigr\|_{1} > t \right] \leq \exp \left( - \frac12 \frac{t^2}{\mathbb{E} \bigl[ \bigl\| \mathbf{X} \bigr\|_{2}^{2} \bigr] + \frac{t}{3} \bigl\| \mathbf{X} \bigr\|_{\infty} } \right).

A similar inequality for negative t is easily obtained by replacing each random variable Xj by its negative. There are also many generalizations, some also known as Bernstein inequalities, and others falling under the general heading of concentration of measure.

2. Lp Norms of Band-Limited Functions

Suppose that f: ℝn → ℂ is band-limited in the sense that its Fourier transform ℱ[f]: ℝn → ℂ is identically zero outside the ball B(0, R) of radius R centred at the origin 0 of ℝn. Suppose also that 1 ≤ p < q < ∞. Then there is a constant C, independent of f, such that

\displaystyle \bigl\| f \bigr\|_{L^{q}(\mathbb{R}^{n})} \leq C R^{n (\frac1p - \frac1q)} \bigl\| f \bigr\|_{L^{p}(\mathbb{R}^{n})} .

In fact, with a possibly different constant, the same inequality holds with a weak Lp quasi-norm on the right-hand side:

\displaystyle \bigl\| f \bigr\|_{L^{q}(\mathbb{R}^{n})} \leq C R^{n (\frac1p - \frac1q)} \bigl\| f \bigr\|_{L^{p,\infty}(\mathbb{R}^{n})} .

This inequality is not too difficult to prove using Young’s inequality for convolutions and standard interpolation between L1,∞ and L2q,∞.

3. Derivatives of Polynomials

Suppose that p: ℂ → ℂ is a polynomial of degree n. Then Bernstein’s inequality for the polynomial p states that, on the unit disc

\displaystyle D := \bigl\{ z \in \mathbb{C} \big| |z| \leq 1 \bigr\},

the degree n of p, multiplied by the maximum absolute volue of p, bounds the maximum absolute value of the derivative of p:

\displaystyle \max_{z \in D} |p'(z)| \leq n \max_{z \in D} |p(z)|.

Similarly, for the kth derivative of p, it holds that

\displaystyle \max_{z \in D} |p^{(k)}(z)| \leq \frac{n!}{(n-k)!} \max_{z \in D} |p(z)|.

This Bernstein inequality is useful in the theory of polynomial approximation, since it gives some measure of approximation of derivatives “for free” once the function itself has been approximated.

A very quick Sobolev embedding

A little while ago, while discussing scaling results in analysis, I had a conversation with a colleague who pointed out to me an elegantly brief proof of the embedding of the critical Sobolev space W1,n(ℝn) into the space BMO(ℝn) of functions of bounded mean oscillation in spatial dimension n. By definition, BMO consists of those L1 functions u such that the semi-norm

\displaystyle \bigl\| u \bigr\|_{\mathrm{BMO}(\mathbb{R}^{n})} := \sup_B \frac1{|B|} \int_B \bigl| u(x) - u_B \bigr| \, \mathrm{d} x

is finite, where the supremum runs over all balls B in ℝn of finite radius, |B| denotes the volume of the ball B, and uB denotes the average of u over B:

\displaystyle u_B := \frac1{|B|} \int_B u(x) \, \mathrm{d} x .

The usual norm on BMO is the sum of the L1 norm and the BMO semi-norm. For p < n, the Sobolev space W1,p(ℝn) embeds into a suitable Lq space; when p > n, the Sobolev space W1,p(ℝn) embeds into a space of Hölder-continuous functions. The space BMO provides a neat (although not complete — see e.g. this paper) description of what happens in the critical case p = n, namely that W1,n(ℝn) embeds into BMO(ℝn); that is, there is a constant c depending only on n such that

\displaystyle \bigl\| u \bigr\|_{\mathrm{BMO}(\mathbb{R}^{n})} \leq c \bigl\| u \bigr\|_{W^{1,n} (\mathbb{R}^n)}.

Continue reading “A very quick Sobolev embedding”