COVID-19 in the UK #3

The chart above (click to zoom) is an updated view of registered deaths in England and Wales according to the ONS up to 4 September, along with data from previous years.

The difference between the red and black lines (highlighted in yellow and labelled A) indicates deaths where COVID-19 was mentioned on the death certificate. The scale of the deaths is sobering, although the worst seems to be over.

There was also a spike in non-COVID-19 deaths (labelled B), which seems to reflect under-treatment of cancer and other serious diseases during the lockdown. The Telegraph expressed concern at this some time ago, and I myself know people in this tragic category. Hopefully, things will be handled better the next time there is an epidemic

Recent deaths (labelled C) have been running slightly below trend, presumably because some of the vulnerable people in the community who would have died about now are already dead.

The bar chart at the bottom shows a year-to-date comparison with previous years. The white additions to the bars for previous years show an adjustment to account for population growth.

Life after solar car: een saaie boel?

A while ago, I posted about the fantastic documentary Driven by Challenges, conceived by Liselotte Kockelkoren-Graas (who was also interviewer, co-director, and executive producer) and produced by D2D Media. The six episodes of this documentary (in Dutch with English subtitles) can be accessed via this playlist. They describe the careers of six former members of Solar Team Eindhoven, which has won the World Solar Challenge Cruiser Class four times with its Stella family of solar cars.

I posted about, and recommended, the documentary as a way of encouraging engineering students to become part of solar car racing and to reap the benefits (once again, watch it here). But there is a downside to building the world’s best Cruiser-class solar car. In comparison, entry-level jobs for engineering graduates may be a little boring – what the Dutch call “een saaie boel.” Evita, in the famous musical of the same name, sings:

High flying, adored, what happens now, where do you go from here?
For someone on top of the world, the view is not exactly clear.
A shame you did it all at twenty-six.
There are no mysteries now,
Nothing can thrill you,
No-one fulfil you.
High flying, adored, I hope you come to terms with boredom.
So famous, so easily, so soon, is not the wisest thing to be.

This can be a genuine problem, but the documentary Driven by Challenges shows us three solutions, and that is what I want to talk about today.

The first solution is to leapfrog up the management ladder. This is an option that is primarily available to graduates who combine engineering talent with people skills. The two examples in the documentary are Wouter van Loon (Escalation Project Lead and Strategic Business Planner at ASML) and Liselotte Kockelkoren-Graas herself (Innovation Lead and Senior R&D Engineer at Vanderlande). It is no accident that both these talented engineers filled people-oriented slots on the solar car team (Wouter handled Sponsorship in 2013, and Liselotte was Account Manager in 2015). Leapfrogging up the management ladder can give you all the challenges you might possibly want.

A quite different solution is to go into Research & Development. An engineering company may not give the coolest projects to fresh graduates, but R&D often lets them do exciting stuff from day one. Two solar car alumni in the documentary (André Snoeck, who handled Finance on the 2013 team, and Patrick Deenen, who was Mech. Eng. System Architect on the 2015 team) went on to do doctorates (doing a doctorate is a lot like working on a solar car, except that it lasts longer, and can sometimes be solitary). There is also another category of R&D worth mentioning: military-related R&D is typically done in specialist government agencies. Such agencies offer fresh graduates a chance to build all kinds of interesting prototypes (and you get to say things like “I could tell you the technical details, but then I’d have to kill you”).

The third solution is to go it alone at a startup. So the world doesn’t fully recognise your talents? Join with like-minded people to start your own company. After all, being part of a world-class solar car team is the best possible practice for that. The documentary includes an interview with Arjo van der Ham, who is co-founder and CTO of Lightyear, a startup company building a commercial solar car.

A fourth solution, not covered in the documentary, is to switch career direction, and to start over in a completely different field. That is less common, but it does sometimes happen. After all, having helped build a solar car is excellent training for about a million different things, and engineering is a way of thinking that is helpful in a wide variety of situations.

Driven by Challenges

Solar Team Eindhoven has won the World Solar Challenge Cruiser Class four times with their Stella family of solar cars (see montage of team photos above). The team’s home base, Eindhoven University of Technology (TU/e) is one of the best technical universities in the world (Times Higher Education rated it #64 in 2017, but that really doesn’t do it justice). So what do engineers do after building the world’s best solar family car?

In 2019, Liselotte Kockelkoren-Graas (account manager of the 2015 team) reconnected with five other former members of Solar Team Eindhoven, in a documentary produced by D2D Media. The six episodes of this beautifully filmed documentary (in Dutch with English subtitles) can be accessed via this playlist.

Liselotte and the other highlighted team alumni are working on some of the coolest engineering projects in the world, in some really cool places (see table and montage below). This list says something about the superb teaching that TU/e offers, the high-tech industry hub in which TU/e is located, and the close links that TU/e (wisely) maintains with its industry neighbours.

Who Now at Where
Liselotte Kockelkoren-Graas (2015 team) Innovation Lead / Senior R&D Engineer, Vanderlande Eindhoven area, NL
Arjo van der Ham (2013 team) Co-Founder / CTO, Lightyear Eindhoven area, NL
André Snoeck (2013 team) Researcher, MIT Megacity Logistics Lab Boston area, US
Patrick Deenen (2015 team) Senior Business Process Analyst, Nexperia & PhD student, TU/e Manchester, UK & Eindhoven, NL
Wouter van Loon (2013 team) Escalation Project Lead / Strategic Business Planner, ASML Taipei, TW & Eindhoven area, NL
Jessie Harms (2017 team) Graduate Intern Eindhoven area, NL & Ahmedabad, IN

Among the things I learned from the documentary Driven by Challenges: exciting things happen when your suitcase vanishes down a belt in an airport, and my beard grows at 5 nanometres per second. It was great to see that the careers of Solar Team Eindhoven alumni are progressing many orders of magnitude faster than that.

All about fixed points

In mathematics, a fixed point of a function f is a value x such that f(x) = x. For example, Wolfram notes that cos(0.7390851332) = 0.7390851332. There need not be such a fixed point, of course, but it is interesting when there is.

Unique fixed points

The Banach fixed-point theorem applies to complete metric spaces – sets where there is a concept of distance and where convergent sequences have a limit.

If f is a function that shrinks things – if, say, the distance between f(x) and f(y) is at most 50%, or 90%, or 99.99999% of the distance between x and y – then the function f will not only have a fixed point, but that fixed point will be unique.

Intuitively, this is clear. You apply the function f to the whole set an infinite number of times, and everything shrinks down to a single point satisfying f(x) = x.

Comic 688 by xkcd, with the fixed point highlighted in pink

It follows that if you stand inside a country, city, or home holding (horizontally) a map of that country, city, or home, then exactly one point in the country, city, or home is exactly underneath the corresponding point on the map. More usefully, this theorem has been used to assign meaning to recursive type expressions in computer science.

Iterated function systems

Iterated function systems are one way of defining fractals. An iterated function system is a family of functions, each of which shrinks things (i.e. is contractive). This family of functions defines a Hutchinson operator on sets of points, and that operator is in turn contractive for a certain definition of distance between sets (see Hutchinson, 1981).

Consequently, that operator has a unique “fixed point,” which is in fact a set of points, like the famous Barnsley fern. The set of points can be (approximately) generated by an iterative process (see code example here):

Least fixed points

As another example of fixed points in action, the Kleene fixed-point theorem assigns meaning to recursively defined computer programs, like these:

  • x = x, a circular definition which is satisfied by infinitely many things, defines x = ⊥, the least defined of all those things
  • z = 0 ⊙ (1 + z), where “” means putting a number in front of a list, defines the infinite list z = 0, 1, 2, 3, 4, 5, …
  • g = function (x) { if (x = 0) then 1 else x * g(x − 1) }, defines g to be the factorial function

Computer programs are modelled by Scott-continuous functions on a partial order where ⊥ is the least element (⊥ means “no information” and is the meaning of any computer program stuck in an infinite loop). Scott-continuous functions may have more than one fixed point, but they have a unique least fixed point, which is the limit of the sequence ⊥ ⊑ f(⊥) ⊑ f(f(⊥)) ⊑ …

This sequence corresponds to what actually happens when the computer attempts to execute recursive definitions like the three in red above.

Brouwer’s fixed point theorem

Even more interesting is Brouwer’s fixed-point theorem. Any continuous function mapping a compact convex set to itself has at least one fixed point.

This is hard to prove in general, though easy to see in the case that the set is simply the interval [0, 1]. We have f(0) − 0 ≥ 0 and f(1) − 1 ≤ 0, so that f(x) − x = 0 for at least one x in the range 0 to 1:

An implication is that, when I stir my cup of coffee, at least one speck of liquid winds up exactly where it started.

Fixed points and dynamics

If the function f represents a state transition function, then a fixed point corresponds to a state that remains the same over time. For example, the logistic mapx (1 − x) has fixed points 0 and 0.75. However, these are both unstable – if you start out close to them, repeated application of the function takes you further away:

  • 0.000001, 0.000004, 0.000016, 0.000064, 0.000256, 0.001024, 0.00409, 0.016295, 0.064117, 0.240023, 0.729649, 0.789046, 0.66581, 0.890028, 0.391511, 0.952921, 0.179452, 0.588995, 0.96832, 0.122706, 0.430598, …
  • 0.749999, 0.750002, 0.749996, 0.750008, 0.749984, 0.750032, 0.749936, 0.750128, 0.749744, 0.750512, 0.748975, 0.752045, 0.745893, 0.758147, 0.733441, 0.782021, 0.681856, 0.867713, 0.459147, 0.993324, 0.026525, …

On the other hand, the logistic map 2.5 x (1 − x) has a stable fixed point (an attractor) at 0.6. Starting at, say, 0.5, repeatedly applying the function gets you closer and closer to 0.6:

  • 0.5, 0.625, 0.585938, 0.606537, 0.596625, 0.601659, 0.599164, 0.600416, 0.599791, 0.600104, 0.599948, 0.600026, 0.599987, 0.600007, 0.599997, 0.600002, 0.599999, 0.6, 0.6, 0.6, 0.6, …

Fixed points are everywhere! Not only that, but the concept is a fruitful one in a diverse collection of fields. For more on fixed points, see this book (available electronically here).