Ebola

The horror that is the Ebola virus outbreak continues to sweep across West Africa. A filovirus (image above) is responsible, and the WHO reports 3,069 cases, with 1,552 deaths. Sadly, there is no sign of the disease slowing down.

The case of two aid workers treated in the USA suggests that the survival rate can be improved with high-quality care, which supports vital functions and counteracts fluid and electrolyte losses. However, delivering such care in Africa can be difficult.

The CDC has more information. One can only hope and pray that medical staff in Africa will be able to halt this terrible disease soon.

The Kuiper Belt

The Kuiper Belt has been in the news lately, because the New Horizons space probe will be visiting it next year. The Kuiper Belt consists of a number of objects on the fringes of the Solar System (from about from about 30 to 55 AU out). Known objects in the belt are green in the image above (image by “WilyD”). Pluto is the best-known of these objects.

The belt is named after the great Dutch-American astronomer Gerard Kuiper (1905–1973), based on his 1951 PNAS article “On the Origin of the Solar System” (although it requires some imagination to see a prediction of the Kuiper Belt in this paper). The photo of Kuiper below is from the Dutch National Archives:

I’m looking forward to pictures from the Kuiper Belt next year!

Complexity vs Randomness

I’ve posted before about randomness and complexity. The montage above illustrates the distinction sometimes made between regular, complex, and random patterns. The text examples below provide another illustration:

Regular Complex Random
aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa … It was the best of times, it was the worst of times, it was the age of wisdom, it was the age of foolishness, it was the epoch of belief, it was the epoch of incredulity, it was the season of Light, it was the season of Darkness, it was the spring of hope, it was the winter of despair, … BdhBt BMgkn YCbfR enFwJ DlMyq KFNoi rRdlu JwdTc IPoim oeFnQ gnhqq pqXon cIVVn rAOrx XtbcQ rZTBF sxeTi hLmBt gREOe Udrwt QHEMW OCBeV gQrHb gKbWa lklBL ivZMg JJrGa xVOZj QQBpb rfZWQ qRKTa ZEktK ofhTD UOXrm ZJAJs LPloV NhFjy …

But how does one measure complexity? Grassberger notes that the widely-used Kolmogorov complexity is simply a measure of randomness. In other words, the use of Kolmogorov complexity implicitly assumes that complexity is just “randomness lite.”

There are three significant groups of people who doubt this. First, those complexity scientists who speak about the “edge of chaos” see the borderlands between regularity and randomness as being critically important, and in need of formal characterisation. However, rather than attempting to measure “complexity” in a way which would give both regular and random patterns low scores, this behavioural zone is now typically studied in terms of correlation length and other critical phenomena.

The second group are those critics of evolution who, believing ex nihilo nihil fit, assert that complexity has to come “from somewhere.” If complexity is just “randomness lite,” then random variation plus natural selection are sufficient to produce complex structures (indeed, in silico, the successes of genetic algorithms and genetic programming support that idea). Doubting this, these critics of evolution (such as Michael Behe) have suggested alternate definitions of complexity (irreducible complexity and specified complexity). However, since these alternate concepts have not been rigorously defined, they are not generally accepted outside the “intelligent design” community.

The third group, which includes figures such as Stuart Kauffman, also claim that random variation plus natural selection is inadequate to explain the evolution of biological complexity. However, they believe that the processes of self-organisation studied by the first group provide the missing explanation. This group does not use an agreed-upon formal definition of complexity, focussing primarily on simulation models in which non-trivial structures emerge. Their approach is interesting, but (as far as I can see) still vigorously debated.

Sleeping in the moonlight… or not


Moritz von Schwind, Selene and Endymion‎, 1831

In 1609, the English writer Thomas Dekker wrote these lines in praise of sleep:

For do but consider what an excellent thing sleep is: it is so inestimable a jewel, that, if a tyrant would give his crown for an hour’s slumber, it cannot be bought … sleep is that golden chain that ties health and our bodies together. Who complains of want, of wounds, of cares, of great men’s oppressions, of captivity, whilst he sleepeth? Beggars in their beds take as much pleasure as kings. Can we therefore surfeit on this delicate ambrosia? Can we drink too much of that, whereof to taste too little tumbles us into a churchyard; and to use it but indifferently throws us into Bedlam? No, no. Look upon Endymion, the Moon’s minion, who slept threescore and fifteen years, and was not a hair the worse for it. Can lying abed till noon then, being not the threescore and fifteen thousandth part of his nap, be hurtful?

The modern prevalence of jet travel and shift work has prompted considerable research in sleep and sleep-related issues, since many travellers and shift-workers struggle to find effective strategies for managing sleep. Thomas Dekker is certainly correct about the effects which sleep deprivation can have – indeed, sleep deprivation can be as debilitating as high blood alcohol concentration.


Full moon (from Weird Tales, Sept 1941)

However, although Selene (the Moon) caused Endymion to sleep, she is unlikely to be of any help here. Past studies have shown that the full moon reduces hours slept (although a recent study finds no effect), and this may underlie traditional beliefs in lunacy caused by the moon… not to mention the legends about lycanthropy.

This post was reblogged and updated from Human Science Explored

Network Topology in Command and Control

I was happy to get my copy of Network Topology in Command and Control: Organization, Operation, and Evolution in the mail today – particularly because it includes my chapter “C2, Networks, and Self-Synchronization” (which rounds off several aspects of my work in this space). There are other interesting chapters too, of course!

Mathematics in action: averaging angles

This post begins a (potential) new series: mathematics in action. Let’s see how it goes.

There are many phenomena in science and engineering which one would like to describe using angles. With a pendulum, for example, one might use 0° to describe the state with the bob at the right, 90° for the bob at the centre moving left, 180° for the bob at the left, 270° for the bob at the centre moving right, and 360° = 0° for the bob at the right again:

This gets even more interesting when we have multiple pendulums (or metronomes, which are just pendulums upside down):

A more practical example of this sort of thing might be a bank of electric generators, which each generate an alternating current as they spin through their cycles:

There are even medical applications (see pages 83–84 of this thesis).

In studying this kind of thing, it is sometimes necessary to find the average of a set of angles. For example: 300°, 330°, 360°, and 30°. This is more complicated than it seems at first sight, since 330° = −30°, 300° = −60°, and so forth.

The solution is to find the average sine: in this case a = −0.2165 (the average of −0.8660, −0.5, 0, and 0.5), and the average cosine: in this case b = 0.8080 (the average of 0.5, 0.8660, 1, and 0.8660).

The average angle is the one whose sine and cosine are a and b, which in this case is: arctan(a/b) = −15° = 345° (Some care is needed in handling the case of negative or zero cosines).

We can define the extent to which the angles are the same by r = sqrt(a2 + b2). This will give r = 0 for angles evenly spaced around a circle, and r = 1 for identical angles. For the set of 300°, 330°, 360°, and 30° (which are quite similar), we get r = 0.8365. To see an example of r defined in this way being used, look at this paper of mine. Having identical or near-identical angles is also important for sets of electric generators.

For those mathematics students that have studied complex numbers, this can all be made a lot easier. Each angle θ is represented as the complex number cos(θ) + i sin(θ) = eiθ. We then average the complex numbers, giving a + b i. Conversion to polar coordinates then gives us r and the average angle φ, since a + b i = r eiφ. This helps to explain why complex numbers are a compulsory part of every engineering course – especially when studying harmonic motion, alternating current, or anything that vibrates.

Paperweights

I find paperweights useful while working, and all the more so if they are appropriate in some way. Here is part of my collection.

On the left is a marble cuboctahedron. Since the edges of this shape form a symmetric graph (all 24 edges are equivalent), it made a nice example while I was working on my 2004 paper “Network Robustness and Graph Topology.” For example, the cuboctahedron is one of the Cayley graphs for the alternating group A4.

The frog on the right, on the other hand, feels right at home with some current work I’m doing on amphibian species distribution modelling.