Adding cooks to the broth

If there are $k$ key ideas needed to produce some important goal (like AI), there is a constant probability per researcher-year to come up with an idea, and the researcher works for $y$ years, what is the the probability of success? And how does it change if we add more researchers to the team?

The most obvious approach is to think of this as y Bernouilli trials with probability p of success, quickly concluding that the number of successes n at the end of y years will be distributed as $\mathrm{Pr}(n)=\binom{y}{n}p^n(1-p)^{y-n}$ . Unfortunately, then the actual answer to the question will be $\mathrm{Pr}(n\geq k) = \sum_{n=k}^y \binom{y}{n}p^n(1-p)^{y-n}$ which is a real mess…

A somewhat cleaner way of thinking of the problem is to go into continuous time, treating it as a homogeneous Poisson process. There is a rate $\lambda$ of good ideas arriving to a researcher, but they can happen at any time. The time between two ideas will be exponentially distributed with parameter $\lambda$ . So the time $t$ until a researcher has $k$ ideas will be the sum of $k$ exponentials, which is a random variable distributed as the Erlang distribution: $f(t; k,\lambda)=\lambda^k t^{k-1} e^{-\lambda t} / (k-1)!$ .

Just like for the discrete case one can make a crude argument that we are likely to succeed if $y$ is bigger than the mean $k/\lambda$ (or $k/p$ ) we will have a good chance of reaching the goal. Unfortunately the variance scales as $k/\lambda^2$ – if the problems are hard, there is a significant risk of being unlucky for a long time. We have to consider the entire distribution.

Unfortunately the cumulative density function in this case is $\mathrm{Pr}(t<y)=1-\sum_{n=0}^{k-1} e^{-\lambda y} (\lambda y)^n / n!$ which is again not very nice for algebraic manipulation. Still, we can plot it easily.

Before we do that, let us add extra researchers. If there are $N$ researchers, equally good, contributing to the idea generation, what is the new rate of ideas per year? Since we have assumed independence and a Poisson process, it just multiplies the rate by a factor of $N$ . So we replace $\lambda$ with $\lambda N$ everywhere and get the desired answer.

This is a plot of the case $k=10, y=10$ .

What we see is that for each number of scientists it is a sigmoid curve: if the discovery probability is too low, there is hardly any chance of success, when it becomes comparable to $k/N$ it rises, and sufficiently above we can be almost certain the project will succeed (the yellow plateau). Conversely, adding extra researchers has decreasing marginal returns when approaching the plateau: they make an already almost certain project even more certain. But they do have increasing marginal returns close to the dark blue “floor”: here the chances of success are small, but extra minds increase them a lot.

We can for example plot the ratio of success probability for $\lambda=0.09$ to the one researcher case as we add researchers:

Even with 10 researchers the success probability is just 40%, but clearly the benefit of adding extra researchers is positive. The curve is not quite exponential; it slackens off and will eventually become a big sigmoid. But the overall lesson seems to hold: if the project is a longshot, adding extra brains makes it roughly exponentially more likely to succeed.

It is also worth recognizing that in this model time is on par with discovery rate and number of researchers: what matters is the product $\lambda y N$ and how it compares to $k$ .

This all assumes that ideas arrive independently, and that there are no overheads for having a large team. In reality these things are far more complex. For example, sometimes you need to have idea 1 or 2 before idea 3 becomes possible: that makes the time $t_3$ of that idea distributed as an exponential plus the distribution of $\mathrm{min}(t_1,t_2)$ . If the first two ideas are independent and exponential with rates $\lambda, \mu$ , then the minimum is distributed as an exponential with rate $\lambda+\mu$ . If they instead require each other, we get a non-exponential distribution (the pdf is $\lambda e^{-\lambda t} + \mu e^{-\mu t} - (\lambda+\mu)e^{-(\lambda+\mu)t}$ ). Some discoveries or bureaucratic scalings may change the rates. One can construct complex trees of intellectual pathways, unfortunately quickly making the distributions impossible to write out (but still easy to run Monte Carlo on). However, as long as the probabilities and the induced correlations small, I think we can linearise and keep the overall guess that extra minds are exponentially better.

In short: if the cooks are unlikely to succeed at making the broth, adding more is a good idea. If they already have a good chance, consider managing them better.

Julia lace gaskets

While surfing the web I came across a neat Julia set, defined by iterating $z_{n+1}=f(z_n)=z_n^2+c/z_n^3$ for some complex constant c. Here are some typical pictures, and two animations: one moving around a circle in the c-plane, one moving slowly down from c=1 to c=0.

The points behind the set

What is going on?

The first step in analysing fractals like this is to find the fixed points and their preimages. $z=\infty$ is clearly mapped to itself. The $z^2$ term will tend to make large magnitude iterates approach infinity, so it is an attractive fixed point.

$z=0$ is a preimage of infinity: iterates falling on zero will be mapped onto infinity. Nearby points will also end up attracted to infinity, so we have a basin of attraction to infinity around the origin. Preimages of the origin will be mapped to infinity in two steps: $0=z^2+c/z^3$ has the solutions $z=(-c)^{1/5}$ – this is where the pentagonal symmetry comes from, since these five points are symmetric. Their preimages and so on will also be mapped to infinity, so we have a hierarchy of basins of attraction sending points away forming some gasket-like structure. The Julia set consists of the points that never gets mapped away, the boundary of this hierarchy of basins.

The other fixed points are defined by $z=z^2+c/z^3$ , which can be rearranged into $z^5-z^4+c=0$ . They don’t have any neat expression and actually do not affect the big picture dynamics as much. The main reason seems to be that they are unstable. However, their location and the derivative close to them affect the shapes in the Julia set as we will see. Their preimages will be surrounded by the same structures (scaled and rotated) as they have.

Below are examples with preimages of zero marked as white circles, fixed points as red crosses, and critical points as black squares.

The set behind the points

A simple way of mapping the dynamics is to look at the (generalized) Mandelbrot set for the function, taking a suitable starting point $z_0=(3c/2)^{1/5}$ and mapping out its fate in the c-plane. Why that particular point? Because it is one of the critical point where $f'(z)=0$ , and a theorem by Julia and Fatou tells us that its fate indicates whether the Julia set is filled or dust-like: bounded orbits of the critical points of a map imply a connected Julia set. When c is in the Mandelbrot set the Julia image has “thick” regions with finite area that do not escape to infinity. When c is outside, then most points end up at infinity, and what remains is either dust or a thin gasket with no area.

The set is much smaller than the vanilla $f(z)=z^2+c$ Mandelbrot, with a cuspy main body surrounded by a net reminiscent of the gaskets in the Julia set. It also has satellite vanilla Mandelbrots, which is not surprising at all: the square term tends to dominate in many locations. As one zooms into the region near the origin a long spar covered in Mandelbrot sets runs towards the origin, surrounded by lacework.

One surprising thing is that the spar does not reach the origin – it stops at $c=4.5 \cdot 10^{-5}$ . Looking at the dynamics, above this point the iterates of the critical point jump around in the interval [0,1], forming a typical Feigenbaum cascade of period doubling as you go out along the spar (just like on the spar of the vanilla Mandelbrot set). But at this location points now are mapped outside the interval, running off to infinity: one of the critical points breaches a basin boundary, causing iterates to run off and the earlier separate basins to merge. Below this point the dynamics is almost completely dominated by the squaring, turning the Julia set into a product of a Cantor set and a circle (a bit wobbly for higher c; it is all very similar to KAM torii). The empty spaces correspond to the regions where preimages of zero throw points to infinity, while along the wobbly circles points get their argument angles multiplied by two for every iteration by the dominant quadratic term: they are basically shift maps. For c=0 it is just the filled unit disk.

So when we allow c to move around a circle as in the animations, the part that passes through the Mandelbrot set has thick regions that thin as we approach the edge of the set. Since the edge is highly convoluted the passage can be quite complex (especially if the circle is “tangent” to it) and the regions undergo complex twisting and implosions/explosions. During the rest of the orbit the preimages just quietly rotate, forming a fractal gasket. A gasket that sometimes looks like a model of the hyperbolic plane, since each preimage has five other preimages, naturally forming an exponential hierarchy that has to be squeezed into a finite roughly circular space.

The Annihilation Score as Satirical Sociology

Today I read The Annihilation Score by Charles Stross during a flight. It is the sixth Laundry novel, and in many ways the weakest. But it might be the intellectually and satirically best.

The Laundry novels are a mix of horror, spy story, geekiness, and satire. This is both a reader-winning combination (transitions from one side of the mixture to another can provide intense contrast, and Stross can give readers a bit of everything) and a balancing problem: each story needs to maintain the right mixture, and the readers often have their own favourite ratios. The Annihilation Score goes further in the direction of satire, reducing the horror and geekiness fairly significantly. This no doubt makes many Laundry fans unhappy. Me too, to some extent: there is nothing more delightful than noticing wordplay based on obscure hermetica and computer science, or the distinctly unsettling implications of thinking through some of the metaphysical assumptions of the setting. However, I think Stross hit on something different in this novel: an important argument disguised as satire.

On the surface the novel suffers from bad pacing: the bulk of it is about management. Not intense action, but rather the issue of how to set up an office, from personnel management to furniture to keeping the funding body happy despite contradictory goals. There is plenty of agency-spotting, with numerous acronymical organisations criss-crossing the story with their interleaved agendas. And finally, in the last fifth, a climactic battle. Typically Laundry novels spend a lot of times establishing a mood and tension for a relatively brief finale where they get unleashed. The Annihilation Score takes this even further, but at least I did not feel much of a build-up. In fact, despite the pressure on the main character she comes across as almost a Westminster Mary Sue: she persists and succeeds at nearly everything, from turning what ought to be a social nightmare into a cozy core team, to handling unseen budgetary constraints.

However, on a deeper level this is not a horror story about inhuman entities from other dimensions threatening to invade our world and their misguided human servants. This is a horror story about the inhuman entity inhabiting Whitehall: government.

Taking jabs at the absurdity, stupidity and inhumanity of bureaucracy has been a staple in the Laundry books. What makes the Annihilation Score stand out is that it actually has a fairly well thought out argument and exposition of why. The basics are familiar from the earlier novels: the iron law of bureaucracy (framed here as the emergent instrumental goal of organisations to preserve themselves), Parkinson’s law, the Snafu principle, empire building, not invented here, in-group out-group dynamics, Something Must Be Done, and so on. The novel does a sociological dive into the internal culture of the subset of bureaucracy dealing with policing. Here there exists a strong ethos about what purpose it actually has, which both serves to recruit and advance people with a compatible mindset and actually maintain some mission focus. Presumably because it would be very noticeable if the police force began too drift too far from its necessary function; compare this with how some branches of academia are kept honest by constant interaction with an unyielding real world, and others diffuse into obscure absurdity since there are only social forces constraining them. But even when a purpose has an apparently clear meaning it can get subtly (or not so subtly) twisted. This is especially true at the top, where the constraints of external practical reality are weakest.

Stross examines the case where bureaucracy recognizes it has an out-of-context problem. Something important yet unknown is intruding, and clearly something must be done to handle it. The problem is of course that following the politician’s syllogism means that whatever fast and decisive action is taken is not going to be based on good knowledge. Worse, if the organisation is centred on dealing with something Very Important like national security it will hence be (1) extremely motivated to do it, (2) discount signals from unimportant (as described by its own value system) organisations or sources. A not so subtle analogy to the Annihilation Score is government handling of many emerging technologies such as encryption. Internal expertise is lacking not just on the technology itself and its full implications, but there is also a lack of expertise in judging the consequences of different actions and expertise in recognizing this kind of expertise.

This is where I think the novel actually succeeds: it plays out a satirical scenario, but the parts are all-too-familiar. Well-meaning people work hard to ensure something agreed to be good, and the result is Moloch. The Sleeper in the Pyramid is not half as scary as the Dweller in Whitehall. Because the later is real.

What is the natural timescale for making a Dyson shell?

KIC 8462852 (“Tabby’s Star”) continues to confuse. I blogged earlier about why I doubt it is a Dyson sphere. SETI observations in radio and optical has not produced any finds. Now there is evidence that it has dimmed over a century timespan, something hard to square with the comet explanation. Phil Plait over at Bad Astronomy has a nice overview of the headscratching.

However, he said something that I strongly disagree with:

Now, again, let me be clear. I am NOT saying aliens here. But, I’d be remiss if I didn’t note that this general fading is sort of what you’d expect if aliens were building a Dyson swarm. As they construct more of the panels orbiting the star, they block more of its light bit by bit, so a distant observer sees the star fade over time.

However, this doesn’t work well either. … Also, blocking that much of the star over a century would mean they’d have to be cranking out solar panels.

Basically, he is saying that a century timescale construction of a Dyson shell is unlikely. Now, since I have argued that we could make a Dyson shell in about 40 years, I disagree. I got into a Twitter debate with Karim Jebari (@KarimJebari) about this, where he also doubted what the natural timescale for Dyson construction is. So here is a slightly longer than Twitter message exposition of my model.

Lower bound

There is a strict lower bound set by how long it takes for the star to produce enough energy to overcome the binding energy of the source bodies (assuming one already have more than enough collector area). This is on the order of days for terrestrial planets, as per Robert Bradbury’s original calculations.

Basic model

Starting with a small system that builds more copies of itself, solar collectors and mining equipment, one can get exponential growth.

A simple way of reasoning: if you have an area $A(t)$ of solar collectors, you will have energy $kA(t)$ to play with, where $k$ is the energy collected per square meter. This will be used to lift and transform matter into more collectors. If we assume this takes $x$ Joules per square meter on average, we get $A'(t) = (k/x)A(t)$ , which makes $A(t)$ is an exponential function with time constant $k/x$ . If a finished Dyson shell has area $A_D\approx 2.8\cdot 10^{23}$ meters and we start with an initial plant of size $A(0)$ (say on the order of a few hundred square meters), then the total time to completion is $t = (x/k)\ln(A_D/A(0))$ seconds. The logarithmic factor is about 50.

If we assume $k \approx 3\cdot 10^2$ W and $x \approx 40.15$ MJ/kg (see numerics below), then t=78 days.

This is very much in line with Robert’s original calculations. He pointed out that given the sun’s power output Earth could be theoretically disassembled in 22 days. In the above calculations the time constant (the time it takes to get 2.7 times as much area) is 37 hours. So for most of the 78 days there is just a small system expanding, not making a significant dent in the planet nor being very visible over interstellar distances; only in the later part of the period will it start to have radical impact.

The timescale is robust to the above assumptions: sun-like main sequence stars have luminosities within an order of magnitude of the sun (so $k$ can only change a factor of 10), using asteroid material (no gravitational binding cost) brings down $x$ by a factor of 10; if the material needs to be vaporized $x$ increases by less than a factor of 10; if a sizeable fraction of the matter is needed for mining/transport/building systems $x$ goes down proportionally; much thinner shells (see below) may give three orders of magnitude smaller $x$ (and hence bump into the hard bound above). So the conclusion is that for this model the natural timescale of terrestrial planetary disassembly into Dyson shells is on the order of months.

Digging into the practicalities of course shows that there are some other issues. Material needs to be transported into place (natural timescale about a year for a moving something 1 AU), the heating effects are going to be major on the planet being disassembled (lots of energy flow there, but of course just boiling it into space and capturing the condensing dust is a pretty good lifting method), the time it takes to convert 1 kg of undifferentiated matter into something useful places a limit of the mass flow per converting device, and so on. This is why our conservative estimate was 40 years for a Mercury-based shell: we assumed a pretty slow transport system.

Numerical values

Estimate for $x$ : assuming that each square meter shell has mass 1 kg, that the energy cost comes from the mean gravitational binding energy of Earth per kg of mass (37.5 MJ/kg), plus processing energy (on the order of 2.65 MJ/kg for heating and melting silicon). Note that using Earth slows things significantly.

I had a conversation with Eric Drexler today, where he pointed out that assuming 1 kg/square meter for the shell is arbitrary. There is a particular area density that is special: given that solar gravity and light pressure both decline with the square of the distance, there exists a particular density $\rho=E/(4 \pi c G M_{sun})\approx 0.78$ gram per square meter, which will just hang there neutrally. Heavier shells will need to orbit to remain where they are, lighter shells need cables or extra weight to not blow away. This might hence be a natural density for shells, making $x$ a factor 1282 smaller.

Linear growth does not work

I think the key implicit assumption in Plait’s thought above is that he imagines some kind of alien factory churning out shell. If it produces it at a constant rate $A'$ , then the time until it a has produced a finished Dyson shell with area $A_D\approx 2.8\cdot 10^{23}$ square meters. That will take $A_D/A'$ seconds.

Current solar cell factories produce on the order of a few hundred MW of solar cells per year; assuming each makes about 2 million square meters per year, we need 140 million billion years. Making a million factories merely brings things down to 140 billion years. To get a century scale dimming time, $A' \approx 8.9\cdot 10^{13}$ square meters per second, about the area of the Atlantic ocean.

This feels absurd. Which is no good reason for discounting the possibility.

Automation makes the absurd normal

As we argued in our paper, the key assumptions are (1) things we can do can be automated, so that if there are more machines doing it (or doing it faster) there will be more done. (2) we have historically been good at doing things already occurring in nature. (3) self-replication and autonomous action occurs in nature. 2+3 suggests exponentially growing technologies are possible where a myriad entities work in parallel, and 1 suggests that this allows functions such as manufacturing to be scaled up as far as the growth goes. As Kardashev pointed out, there is no reason to think there is any particular size scale for the activities of a civilization except as set by resources and communication.

Incidentally, automation is also why cost overruns or lack of will may not matter so much for this kind of megascale projects. The reason Intel and AMD can reliably make billions of processors containing billions of transistors each is that everything is automated. Making the blueprint and fab pipeline is highly complex and requires an impressive degree of skill (this is where most overruns and delays happen), but once it is done production can just go on indefinitely. The same thing is true of Dyson-making replicators. The first one may be a tough problem that takes time to achieve, but once it is up and running it is autonomous and merely requires some degree of watching (make sure it only picks apart the planets you don’t want!) There is no requirement of continued interest in its operations to keep them going.

Likely growth rates

But is exponential growth limited mostly by energy the natural growth rate? As Karim and others have suggested, maybe the aliens are lazy or taking their time? Or, conversely, that multi century projects are unexpectedly long-term and hence rare.

Obviously projects could occur with any possible speed: if something can construct something in time X, it can in generally be done half as fast. And if you can construct something of size X, you can do half of it. But not every speed or boundary is natural. We do not answer the question of why a forest or the Great Barrier reef have the size they do by cost overruns stopping them, or that they will eventually grow to arbitrary size, but the growth rate is so small that it is imperceptible. The spread of a wildfire is largely set by physical factors, and a static wildfire will soon approach its maximum allowed speed since part of the fire that do not spread will be overtaken by parts that do. The same is true for species colonizing new ecological niches or businesses finding new markets. They can run slow, it is just that typically they seem to move as fast as they can.

Human economic growth has been on the order of 2% per year for very long historical periods. That implies a time constant $\ln(1.02)\approx 50$ years. This is a “stylized fact” that remained roughly true despite very different technologies, cultures, attempts at boosting it, etc. It seems to be “natural” for human economies. So were a Dyson shell built as a part of a human economy, we might expect it to be completed in 250 years.

What about biological reproduction rates? Merkle and Freitas lists the replication time for various organisms and machines. They cover almost 25 orders of magnitude, but seem to roughly scale as $\tau \approx c M^{1/4}$ , where $M$ is the mass and $c\approx 10^7$ . So if a total mass $M_T$ needs to be converted into replicators of mass $M$ , it will take time $t=\tau\ln(M_T)/\ln(2)$ . Plugging in the first formula gives $t=c M^{1/4} \ln(M_T)/\ln(2)$ . The smallest independent replicators have $M_s=10^{-15} kg$ (this gives $\tau_s=10^{3.25}=29$ minutes) while a big factory-like replicator (or a tree!) would have $M_b=10^5$ ( $\tau_b=10^{8.25}=5.6$ years). In turn, if we set $M_T=A_D\rho=2.18\cdot 10^{20}$ (a “light” Dyson shell) the time till construction ranges from 32 hours for the tiny to 378 years for the heavy replicator. Setting $M_T$ to an Earth mass gives a range from 36 hours to 408 years.

The lower end is infeasible, since this model assumes enough input material and energy – the explosive growth of bacteria-like replicators is not possible if there is not enough energy to lift matter out of gravity wells. But it is telling that the upper end of the range is merely multi-century. This makes a century dimming actually reasonable if we think we are seeing the last stages (remember, most of the construction time the star will be looking totally normal); however, as I argued in my previous post, the likelihood of seeing this period in a random star being englobed is rather low. So if you want to claim it takes millennia or more to build a Dyson shell, you need to assume replicators that are very large and heavy.

[Also note that some of the technological systems discussed in Merkle & Freitas are significantly faster than the main branch. Also, this discussion has talked about general replicators able to make all their parts: if subsystems specialize they can become significantly faster than more general constructors. Hence we have reason to think that the upper end is conservative.]

Conclusion

There is a lower limit on how fast a Dyson shell can be built, which is likely on the order of hours for manufacturing and a year of dispersion. Replicator sizes smaller than a hundred tons imply a construction time at most a few centuries. This range includes the effect of existing biological and economic growth rates. We hence have a good reason to think most Dyson construction is fast compared to astronomical time, and that catching a star being englobed is pretty unlikely.

I think that models involving slowly growing Dyson spheres require more motivation than models where they are closer to the limits of growth.

A prime minimal surface

A short while ago I mentioned lacunary functions, complex functions that are analytic inside a disc but cannot be continued outside it. Then I remembered the wonderful Weierstrass-Enneper representation formula, which assigns a minimal surface to (nearly) any pair of complex functions. What happens when you make a minimal surface from a lacunary function?

I was not first with this idea. In fact, F.F. de Brito used this back in 1992 to demonstrate that there exist complete embedded minimal surfaces in 3-space that are contained between two planes.

Here is the surface defined by the function $g(z)=\sum_{p \mathrm{is prime}} z^p$ , the Taylor series that only includes all prime powers, combined with $f(z)=1$ .

Close to zero, the surface is flat. Away from zero it begins to wobble as increasingly high powers in the series begin to dominate. It behaves very much like a higher-degree Enneper surface, but with a wobble that is composed of smaller wobbles. It is cool to consider that this apparently irregular pattern corresponds to the apparently irregular pattern of all primes.

Predictions for 2016

The ever readable Slate Star Codex has a post about checking how accurate the predictions for 2015 were; overall Scott Alexander seems pretty well calibrated. Being a born follower I decided to make a bunch of predictions to check my calibration in a year’s time.

Here is my list of predictions, with my confidence (some predictions obviously stolen):

No nuclear war: 99%
No terrorist attack in the USA will kill > 100 people: 95%
I will be involved in at least one published/accepted-to-publish research paper by the end of 2015: 95%
Vesuvius will not have a major eruption: 95%
I will remain at my same job through the end of 2015: 90%
MAX IV in Lund delivers X-rays: 90%
Andart II will remain active: 90%
Israel will not get in a large-scale war (ie >100 Israeli deaths) with any Arab state: 90%
US will not get involved in any new major war with death toll of > 100 US soldiers: 90%
New Zeeland has not decided to change current flag at end of year: 85%
No multi-country Ebola outbreak: 80%
Assad will remain President of Syria: 80%
ISIS will control less territory than it does right now: 80%
North Korea’s government will survive the year without large civil war/revolt: 80%
The US NSABB will allow gain of function funding: 80%
US presidential election: democratic win: 75%
A general election will be held in Spain: 75%
Syria’s civil war will not end this year: 75%
There will be no NEO with Torino Scale >0 on 31 Dec 2016: 75%
The Atlantic basin ACE will be below 96.2: 70%
Sweden does not get a seat on the UN Security Council: 70%
Bitcoin will end the year higher than $200: 70%
Another major eurozone crisis: 70%
Brent crude oil will end the year lower than $60 a barrel: 70%
I will actually apply for a UK citizenship: 65%
UK referendum votes to stay in EU: 65%
China will have a GDP growth above 5%: 65%
Evidence for supersymmetry: 60%
UK larger GDP than France: 60%
France GDP growth rate less than 2%: 60%
I will have made significant progress (4+ chapters) on my book: 55%
Iran nuclear deal holding: 50%
Apple buys Tesla: 50%
The Nikkei index ends up above 20,000: 50%

The point is to have enough that we can see how my calibration works.

Looking for topics leads to amusing finds like the predictions of Nostradamus for 2015. Given that language barriers remain, the dead remain dead, lifespans are less than 200, there has not been a Big One in western US nor has Vesuvius erupted, and taxes still remain, I think we can conclude he was wrong or the ability to interpret him accurately is near zero. Which of course makes his quatrains equally useless.

Some math for Epiphany

Analytic functions helping you out

Recently I chatted with a mathematician friend about generating functions in combinatorics. Normally they are treated as a neat symbolic trick: you have a sequence $a_n$ (typically how many there are of some kind of object of size $n$ ), you formally define a function $f(z)=\sum_{n=0}^\infty a_n z^n$ , you derive some constraints on the function, and from this you get a formula for the $a_n$ or other useful data. Convergence does not matter, since this is purely symbolic. We used this in our paper counting tie knots. It is a delightful way of solving recurrence relations or bundle up moments of probability distributions.

I innocently wondered if the function (especially its zeroes and poles) held any interesting information. My friend told me that there was analytic combinatorics: you can actually take $f(z)$ seriously as a (complex) function and use the powerful machinery of complex analysis to calculate asymptotic behavior for the $a_n$ from the location and type of the “dominant” singularities. He pointed me at the excellent course notes from a course at Princeton linked to the textbook by Philippe Flajolet and Robert Sedgewick. They show a procedure for taking combinatorial objects, converting them symbolically into generating functions, and then get their asymptotic behavior from the properties of the functions. This is extraordinarily neat, both in terms of efficiency and in linking different branches of math.

Plot of z/(1-z-z^2), the generating function of the Fibonacci numbers. It has poles at (1+sqrt(5))/2 (the dominant pole giving the overall asymptotic growth of Fibonacci numbers) and (1-sqrt(5))/2, which does not contribute much to the asymptotic behavior.

In our case, one can show nearly by inspection that the number of Fink-Mao tie knots grow with the number of moves as $\sim 2^n$ , while single tuck tie knots grow as $\sim \sqrt{6}^n$ .

Analytic functions behaving badly

The second piece of math I found this weekend was about random Taylor series and lacunary functions.

If $f(z)=\sum_{n=0}^\infty X_n z^n$ where $X_n$ are independent random numbers, what kind of functions do we get? Trying it with complex Gaussian $X_n$ produces a disk of convergence with some nondescript function on the inside.

Plot of function with a Gaussian Taylor series. Color corresponds to stereographic mapping of the complex plane to a sphere, with infinity being white and zeros black. The domain of convergence is the unit circle.

Replacing the complex Gaussian with a real one, or uniform random numbers, or even power-law numbers gives the same behavior. They all seem to have radius 1. This is not just a vanilla disk of convergence (where an analytic function reaches a pole or singularity somewhere on the boundary but is otherwise fine and continuable), but a natural boundary – that is, a boundary so dense with poles or singularities that continuation beyond it is not possible at all.

The locus classicus about random Taylor series is apparently Kahane, J.-P. (1985), Some Random Series of Functions. 2nd ed., Cambridge University Press, Cambridge.

A naive handwave argument is that for $|z|<1$ we have an exponentially decaying sequence of $z^n$ , so if the $X_n$ have some finite average size $E(X)$ and not too divergent variance we should expect convergence, while outside the unit circle any nonzero $E(X)$ will allow it to diverge. We can even invoke the Markov inequality $P(X>t) \leq E(X)/t$ to argue that a series $\sum X_n f(n)$ would converge if $\sum f(n)/n$ converges. However, this is not correct enough for proper mathematics. One entirely possible Gaussian outcome is $X_n=1,10,100,1000,\ldots$ or worse. We need to speak of probabilistic convergence.

Andrés E. Caicedo has a good post about how to approach it properly. The “trick” is the awesome Kolmogorov zero-one law that implies that since the radius of convergence depends on the entire series X_n rather than any finite subset (and they are all independent) it will be a constant.

This kind of natural boundary disk of convergence may look odd to beginning students of complex analysis: after all, none of the functions we normally encounter behave like this. Except that this is of course selection bias. If you look at the example series for lacunary functions they all look like fairly reasonable sparse Taylor series like $z+z^4+z^8+z^16+^32+\lddots$. In calculus we are used to worrying that the coefficients in front of the z-terms of a series don’t diminish fast enough: having fewer nonzero terms seems entirely innocuous. But as Hadamard showed, it is enough that the size of the gaps grow geometrically for the function to get a natural boundary (in fact, even denser series do this – for example having just prime powers). The same is true for Fourier series. Weierstrass’ famous continuous but nowhere differentiable function is lacunary (in his 1880 paper on analytic continuation he gives the example $\sum a_n z^{b^n}$ of an uncontinuable function). In fact, as Emile Borel found and Steinhardt eventually proved in a stricter sense, in general (“almost surely”) a Taylor series isn’t continuable because of boundaries.

The function sum_p z^p, where p runs over the primes. — The function [latex]sum_p z^p[/latex], where [latex]p[/latex] runs over the primes.

One could of course try to combine the analytic combinatorics with the lacunary stuff. In a sense a lacunary generating function is a worst case scenario for the singularity-measuring methods used in analytical combinatorics since you get an infinite number of them at a finite and equal distance, and now have to average them together somehow. Intuitively this case seems to correspond to counting something that becomes rarer at a geometric rate or faster. But the Borel-Steinhardt results suggest that even objects that do not become rare could have nasty natural boundaries – if the number $a_n$ were due to something close enough to random we should expect estimating asymptotics to be hard. The funniest example I can think of is the number of roots of Chaitin-style Diophantine equations where for each $n$ it is an independent arithmetic fact whether there are any: this is hardcore random, and presumably the exact asymptotic growth rate will be uncomputable both practically and theoretically.

Big picture thinking

In Michaelmas term 2015 we ran a seminar series on Big Picture Thinking at FHI. The videos of most seminars are online.

I gave a talk on observer selection effects, and here are my somewhat overdone lecture notes. Covers selection bias, anthropic reasoning, anthropic shadows, nuclear war near misses, physics disasters, the Doomsday Argument, the Fermi Paradox, the Simulation Argument, fine tuning and Boltzmann brains.

Starkiller base versus the ideal gas law

My friend Stuart explains why the Death Stars and the Starkiller Base in the Star Wars universe are inefficient ways of taking over the galaxy. I generally agree: even a super-inefficient robot army will win if you simply bury enemy planets in robots.

But thinking about the physics of absurd superweapons is fun and warms the heart.

The ideal gas law: how do you compress stars?

My biggest problem with the Starkiller Base is the ideal gas law. The weapon works by sucking up a star and then beaming its energy or plasma at remote targets. A sun-like star has a volume around 1.4*10¹⁸ cubic kilometres, while an Earthlike planet has a volume around 10¹² cubic kilometres. So if you suck up a star it will get compressed by a factor of 1.4 million times. The ideal gas law states that pressure times volume equals temperature times the number of particles and some constant: $PV=nRT$

1.4 million times less volume needs to be balanced somehow: either the pressure P has to go down, the temperature T has to go up, or the number of particles n need to go down.

Pressure reduction seems to be a non-starter, unless the Starkiller base actually contains some kind of alternate dimension where there is no pressure (or an enormous volume).

The second case implies a temperature increase by a factor of a 1.4 million. Remember how hot a bike pump gets when compressing air: this is the same effect. This would heat the photosphere gas to 8.4 billion degrees and the core to 2.2*10¹³ K, 22 TeraKelvin; the average would be somewhere between, on the hotter side. We are talking about temperatures microseconds after the Big Bang, hotter than a supernova: protons and neutrons melt at 0.5–1.2 TK into a quark-gluon plasma. Excellent doomsday weapon material but now containment seems problematic. Even if we have antigravity forcefields to hold the star, the black-body radiation is beyond the supernova range. Keeping it inside a planet would be tough: the amount of neutrino radiation would likely blow up the surface like a supernova bounce does.

Maybe the extra energy is bled off somehow? That might be a way to merely get super-hot plasma rather than something evaporating the system. Maybe those pesky neutrinos can be shunted into hyperspace, taking most of the heat with them (neutrino cooling can be surprisingly fast for very hot objects; at these absurd temperatures it is likely subsecond down to mere supernova temperatures).

Another bizarre and fun approach is to reduce the number of gas particles: simply fuse them all into a single nucleus. A neutron star is in a sense a single atomic nucleus. As a bonus, the star would now be a tiny multikilometre sphere held together by its own gravity. If n is reduced by a factor of 10⁵⁷ it could outweigh the compression temperature boost. There would be heating from all the fusion; my guesstimate is that it is about a percent of the mass energy, or 2.7*10⁴⁵ J. This would heat the initial gas to around 96 billion degrees, still manageable by the dramatic particle number reduction. This approach still would involve handling massive neutrino emissions, since the neutronium would still be pretty hot.

In this case the star would remain gravitationally bound into a small blob: convenient as a bullet. Maybe the red “beam” is actually just an accelerated neutron star, leaking mass along its trajectory. The actual colour would of course be more like blinding white with a peak in the gamma ray spectrum. Given the intense magnetic fields locked into neutron stars, moving them electromagnetically looks pretty feasible… assuming you have something on the other end of the electromagnetic field that is heavier or more robust. If a planet shoots a star-mass bullet at a high velocity, then we should expect the recoil to send the planet moving at about a million times faster in the opposite direction.

Other issues

We have also ignored gravity: putting a sun-mass inside an Earth-radius means we get 333,000 times higher gravity. We can try to hand-wave this by arguing that the antigravity used to control the star eating also compensates for the extra gravity. But even a minor glitch in the field would produce an instant, dramatic squishing. Messing up the system* containing the star would not produce conveniently dramatic earthquakes and rifts, but rather near-instant compression into degenerate matter.

(* System – singular. Wow. After two disasters due to single-point catastrophic failures one would imagine designers learning their lesson. Three times is enemy action: if I were the Supreme Leader I would seriously check if the lead designer happens to be named Skywalker.)

There is also the issue of the amount of energy needed to run the base. Sucking up a star from a distance requires supplying the material with the gravitational binding energy of the star, 6.87*10⁴¹ J for the sun. Doing this over an hour or so is a pretty impressive power, about 1.9*10³⁸ W. This is about 486 billion times the solar luminosity. In fact, just beaming that power at a target using any part of the electromagnetic spectrum would fry just about anything.

Of course, a device that can suck up a star ought to be able to suck up planets a million times faster. So there is no real need to go for stars: just suck up the Republic. Since the base can suck up space fleets too, local defences are not much of a problem. Yes, you may have to go there with your base, but if the Death Star can move, the Starkiller can too. If nothing else, it could use its beam to propel itself.

If the First Order want me to consult on their next (undoubtedly even more ambitious) project I am open for offers. However, one iron-clad condition given recent history is that I get to work from home, as far away as possible from the superweapon. Ideally in a galaxy far, far away.

Did amphetamines help Erdős?

During my work on the Paris talk I began to wonder whether Paul Erdős (who I used as an example of a respected academic who used cognitive enhancers) could actually have been shown to have benefited from his amphetamine use, which began in 1971 according to Hill (2004). One way of investigating is his publication record: how many papers did he produce per year before or after 1971? Here is a plot, based on Jerrold Grossman’s 2010 bibliography:

Productivity of Paul Erdos over his life. Green dashed line: amphetamine use, red dashed line: death. Crosses mark named concepts.

The green dashed line is the start of amphetamine use, and the red dashed life is the date of death. Yes, there is a fairly significant posthumous tail: old mathematicians never die, they just asymptote towards zero. Overall, the later part is more productive per year than the early part (before 1971 the mean and standard deviation was 14.6±7.5, after 24.4±16.1; a Kruskal-Wallis test rejects that they are the same distribution, p=2.2e-10).

This does not prove anything. After all, his academic network was growing and he moved from topic to topic, so we cannot prove any causal effect of the amphetamine: for all we know, it might have been holding him back.

One possible argument might be that he did not do his best work on amphetamine. To check this, I took the Wikipedia article that lists things named after Erd ő s, and tried to find years for the discovery/conjecture. These are marked with red crosses in the diagram, slightly jittered. We can see a few clusters that may correspond to creative periods: one in 35-41, one in 46-51, one in 56-60. After 1970 the distribution was more even and sparse. 76% of the most famous results were done before 1971; given that this is 60% of the entire career it does not look that unlikely to be due to chance (a binomial test gives p=0.06).

Again this does not prove anything. Maybe mathematics really is a young man’s game, and we should expect key results early. There may also have been more time to recognize and name results from the earlier career.

In the end, this is merely a statistical anecdote. It does show that one can be a productive, well-renowned (if eccentric) academic while on enhancers for a long time. But given the N=1, firm conclusions or advice are hard to draw.

Erdős’s friends worried about his drug use, and in 1979 Graham bet Erdős $500 that he couldn’t stop taking amphetamines for a month. Erdős accepted, and went cold turkey for a complete month. Erdős’s comment at the end of the month was “You’ve showed me I’m not an addict. But I didn’t get any work done. I’d get up in the morning and stare at a blank piece of paper. I’d have no ideas, just like an ordinary person. You’ve set mathematics back a month.” He then immediately started taking amphetamines again. (Hill 2004)

Andart II

Part of Anders' Exoself

Author: admin