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In conversation with Bernard Silverman

It’s been said that a degree in mathematics opens many doors, but to many this might seem a slight exaggeration. Bernard Silverman, however, is an excellent example of a mathematics graduate who has indeed done it all. Silverman is currently the chief scientific advisor to the Home Office, a statistician, and an Anglican priest. These are just a few examples of his many achievements, starting from the gold medal he won at the 1970 International Mathematics Olympiad—the only person to do so from the western side of the iron curtain—at the beginning of his mathematical career. He went on to read mathematics at university, and eventually obtained a PhD in data analysis in 1977. “I was always interested in maths, but as time went on I became keen on doing it in a way that has applications in different things, and that is what drew me to statistics.” He jokingly adds that he felt he was never good enough to be a pure mathematician. In the course of our conversation with him, he took us on a journey through the diverse areas in which he has applied his statistical approach. Continue reading

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Linear algebra… with diagrams

A succinct—if somewhat reductive—description of linear algebra is that it is the study of vector spaces over a field, and the associated structure-preserving maps known as linear transformations. These concepts are by now so standard that they are practically fossilised, appearing unchanged in textbooks for the best part of a century.

While modern mathematics has moved to more abstract pastures, the theorems of linear algebra are behind a surprising number of world-changing technologies: from quantum computing and quantum information, through control and systems theory, to big data and machine learning. All rely on various kinds of circuit diagrams, eg electrical circuits, quantum circuits or signal flow graphs. Circuits are geometric/topological entities, but have a vital connection to (linear) algebra, where the calculations are usually carried out.

In this article, we cut out the middle man and rediscover linear algebra itself as an algebra of circuit diagrams. The result is called graphical linear algebra and, instead of using traditional definitions, we will draw lots of pictures. Mathematicians often get nervous when given pictures, but relax: these ones are rigorous enough to replace formulas.

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Debugging insect dynamics

Social dynamics are complex and have evolved over many generations. One strategy that is used is that of altruism: the act of helping someone else at a cost to yourself. In some insects, this takes on an extreme case where workers sacrifice their own fertility to help raise the queen’s eggs instead. While this may seem to go against the idea that animals want to pass on as many of their genes as possible, we’ll see why this is actually a viable strategy.

Game theory examines how the frequencies of different strategies played in a game change over time. Evolutionary game theory looks at the special case where players cannot change the strategy they play: they’re stuck with the strategy they’ve inherited from the previous generation. In each round, players are randomly matched up and play a game, leading to an outcome dependent on their strategies. This outcome is called a payoff, and it affects their fitness and thus the number of offspring they produce. In the 1970s, John Maynard Smith and George Price developed evolutionary game theory to investigate ritualistic fighting behaviour in animals. A good example is how male stags will compete for territory during the mating season. Physical contact is actually unlikely to occur and the stags can spend hours staring and roaring at each other to determine who is the strongest. If things do escalate, many species of stags have branched antlers allowing them to wrestle rather than impale each other. Species with straighter antlers will tend not to use them in fights, but will resort to biting and kicking, which is far less dangerous. This strategy of assessing your opponent first and picking your fights carefully is clearly beneficial for the species as a whole, but it wasn’t originally clear why an aggressive strategy (where you kill your opponent and pass on your genes) isn’t more common in the animal kingdom.

Smith and Price sought to examine this and they devised the hawk–dove game. A population (of the same species) is split into two groups: hawks and doves. Hawks are aggressive and will play until they win or are seriously injured. On the other hand, a dove is a pacifist and will surrender if its opponent gets aggressive (so it will never get injured). In this game, two players are matched up and compete for a resource (eg food), and the outcome depends on the strategies they play. These dynamics are shown in the following payoff matrix:

meets hawk meets dove
If hawk $(v-c)/2$ $v$
If dove $0$ $v/2$

Here, $v$ is the value of the resource, while $c$ is the cost of injury (from a hawk losing to another hawk). Typically in nature, we find that $c$ is much larger than $v$. To explain the entries of the payoff matrix: when a hawk meets another hawk, there is a 50% chance it will win, gaining $v$; but a 50% chance it will lose, losing $c$. When a hawk meets a dove, it will always win, gaining $v$, while the dove always loses without injury, receiving 0. When a dove meets another dove, there is a 50% chance it will win, gaining $v$ and, if it loses, it does not get injured but receives 0.

These dynamics can be analysed by the replicator equations. The change in proportion of a strategy $i$ ($x_i$) is given by the fitness of the strategy ($f_i$), minus the average fitness of the population, all multiplied by the proportion of strategy $i$: \[
\frac{\mathrm{d}x_i}{\mathrm{d}t}=x_i\Big(f_i(\mathbf{x})- \sum_{j=1}^n x_j\hspace{2pt} f_j(\mathbf{x}) \Big).\]
The distribution of the population into the $n$ strategies is given by the vector $\mathbf{x}=(x_1,\ldots,x_n)$ which, since they are proportions, has entries summing to 1. Using the values from the payoff matrix above leads to a single differential equation (since $x_2=1-x_1$), with a globally stable steady state where the proportion of hawks is $v/c$, which is closer to 0 than 1. While this can explain the lack of aggressive strategies seen in nature, an extension of this is the hawk–dove–assessor game.

An assessor plays as a hawk if they are stronger than their opponent, and as a dove if they are weaker. This is precisely what we tend to see in ritualistic fights in nature. The payoff matrix is given below, which you can verify yourself.

meets hawk meets dove meets assessor
If hawk $(v-c)/2$ $v$ $(v-c)/2$
If dove $0$ $v/2$ $v/4$
If assessor $v/2$ $3v/4$ $v/2$

We find that the strategy of being an assessor is an evolutionary stable strategy. This means that if the entire population is playing as assessors, then any invasion by another strategy cannot succeed and spread. The assessors will always dominate eventually.

Many different strategies can be represented in game theory, including cooperation, spite, selfishness and altruism. Altruism is when a player does something beneficial to the recipient at a cost to itself. This can be seen in games with a memory or reputation system: if the cost is low but the benefit is high, then an individual may act altruistically in the hope of the other player returning the favour later on, leading to a net benefit for them both. Perhaps the most obvious reason for an altruistic act, though, is if the players are related, meaning that they have genes in common. This stems from the idea of inclusive fitness:\begin{align}\text{inclusive fitness}&=\text{individual fitness}+(\text{relatedness}\times\text{relative’s individual fitness}),\\
\omega_i&=f_i+\sum_{j \neq i} R_{ij}\hspace{2pt} f_j.\end{align}
This means that when examining the dynamics of gene frequencies, the fitness of an individual’s family should also play a role since some of the genes will be shared. Relatedness is defined to be the probability that a gene picked randomly from each individual at the same locus (position) is identical by descent. This works out intuitively: 0.5 between you and a sibling/parent/son/daughter (since each parent gives half of their genes to their offspring), 0.25 between you and an uncle/aunt, 0.125 between you and a cousin. Another name for inclusive fitness theory is selfish gene theory, popularised by Richard Dawkins’ book The Selfish Gene, which was influenced by ideas from fellow biologist George Williams. The term ‘selfish’ refers to how some genes may prioritise their own survival (over many generations) over that of the individual or even species. This gene-centred view of evolution helps explain altruism.

William Donald “Bill” Hamilton was an evolutionary biologist who completed his PhD at the London School of Economics and University College London. He claimed that altruistic acts are favoured (and as a strategy, can spread through the population) if the relatedness between the players is greater than the cost to benefit ratio of the act, ie \[R>\frac{C}{B}.\]
This became known as Hamilton’s rule and, while it can be hard to quantify and test, a simple example occurs in prairie dogs. When these rodents are above ground and spot a predator, an individual is more likely to sound an alarm call when relatives are close by: an action that is costly as the individual draws attention to itself.

Colony bee-hive-iour

Atta cephalotes castes (Gamekeeper, CC BY-SA 3.0)

An extreme case of altruism is eusociality:  the highest level of organisation in animal social structure. This structure is what you probably think of when you consider an insect colony: a queen laying eggs and thousands of workers maintaining the nest. Some other traits of this include cooperative brood care and overlapping generations; however, its most distinct trait is the division of labour into castes. These castes are specialised in that individuals from one caste lose the ability to perform the tasks of individuals from another caste. These castes don’t just vary physically, they can express different behaviours and even instincts! In the figure on the previous page, we can see many types of workers on the left, a soldier in the middle and two queens on the right. The soldier is larger than the workers and has a stronger jaw—interestingly this is mainly used for foraging heavy objects and not to defend the nest: that job is up to the workers. Most eusocial animals can be found in the third largest order of insects, called Hymenoptera, which includes bees, wasps and ants.

Hymenopterans are also haplodiploids, ie males have one set of chromosomes (haploids) and females have the usual two (diploids). This bizarre fact means that males actually hatch from unfertilised eggs and females from fertilised eggs. Some strange relationships can come from this. For example, males have no father or sons, but have a grandfather and can have grandsons! But the most intriguing fact is that the workers (who are all female and make up the vast majority of a colony) help raise new brothers and sisters produced by the queen instead of having their own offspring. This fact puzzled Charles Darwin, as it wasn’t clear how a trait that leads to an individual not reproducing and passing on their own genes can be so prevalent in a population. This would mean that their individual fitness is zero; however, going back to inclusive fitness theory, we can see that they can still benefit from their relatives. In fact, we’ll see that this benefit outweighs that of having their own offspring.

Fig. 1: Family tree of haplodiploid insects


A mated queen can produce fertilised and unfertilised eggs for the rest of her life. From the figure to the left, we can see that female workers are more related to their sisters than any other relative (including possible offspring). Explicitly, two female workers share exactly the same set of blue chromosomes (from their father), which is already a relatedness of $0.5$. The other set (in red) can either be exactly the same or different, depending on which was inherited from the queen, giving on average another $0.25$, and making their relatedness $R=0.75$. Following Hamilton’s rule, it indeed makes sense to help raise new sisters instead of offspring, who only have a relatedness of $0.5$. In most species of eusocial Hymenoptera, the queen is aggressive and releases pheromones to discourage workers from laying eggs and, in some cases, from even ovulating. However, it seems that not all workers are happy with this arrangement, and this can lead to several different types of conflict.

Conflict: sex allocation

The queen is equally related to male and female offspring (with a ratio of 1:1) but female workers prefer sisters to brothers (with a ratio of 3:1). Consequently, it is common for the female workers to eat or kill male larvae so that the colony’s resources are not used in raising them. This means that the queen has wasted energy producing and laying the male eggs. A common solution to this is to produce a brood of all the same sex, reducing the motivation for the males to be destroyed. In reality, the ratio is between the two ideals for the queen and female workers. For most of these species, males (AKA drones) do not help with work in the colony: their only role is to mate with a young queen—so, apart from the genetics, female workers may benefit even less from having brothers. For the queen, however, males are still important to help spread her genes even further: if the males are able to mate with a new queen who then starts her own colony, all of the new workers will carry one set of chromosomes from the original queen. In some species, it has been noticed that the queen will lay batches of male eggs when food supplies are low, as female workers’ fitnesses are more affected by food during their developmental stages. However, since males don’t even help with foraging for food, this strategy would only work in the short term.

Conflict: male rearing

Apis mellifera: queen and workers (Jessica Lawrence, CC BY 3.0)

In honey bees, 7% of male eggs are from workers but only 0.1% of adult males are a worker’s son. Why would workers lay eggs in the first place? Well, they are more related to their own offspring ($R=0.5$) than to any brothers ($R=0.25$) that the queen produces. However, laying workers are less hard-working and the queen would rather spend resources raising new children than grandchildren (to whom she is less related), so she tries to prevent this using pheromones. These pheromones inhibit the workers’ ability to lay eggs but are less effective in large colonies, especially if the queen is old.

Worker policing occurs through workers destroying other worker-laid eggs. We can calculate from the family tree (Fig. 1) that workers are more related to new sisters ($R=0.75$) than any nephews ($R=0.33$). The calculation for the second relation works as follows: comparing a female worker to her nephew, there are three possible cases involving the three different sets of chromosomes in the family tree (the blue set of chromosomes from the queen’s mate and the two red sets of chromosomes from the queen herself). One case is where the nephew has the blue set of chromosomes, which his aunt will also have, meaning she shares 50% of her genes with him. If the nephew has a red set of chromosomes, they can either be the same set as his aunt’s (again, 50% of her genes will be shared), or they can be the ‘other’ red set originally from the queen (so his aunt will share none of her genes in this case). Thus on average, a worker has a relatedness of $R=(0.5+0.5+0)/3$ to her nephew. Workers can determine whom the eggs belong to through chemical markers. You may think that there is a selection pressure (in evolution) towards workers who can lay eggs that mimic the queen’s… and you’d be right! These are called ‘anarchic workers’ and in the Cape honey bee, this trait is affected by multiple genes; but, for various reasons, the cost of having these altered genes outweighs their benefit. Note that this conflict does not appear in every eusocial hymenopteran: in some species, the castes are so specialised that workers aren’t even capable of reproducing in the first place.

Conflict: caste fate

Apis mellifera pupae (Waugsberg, CC BY-SA 3.0)

Why wouldn’t a worker choose to develop as a queen instead? Doing so would be selfish but it yields a much greater inclusive fitness. In many species, the determining factor as to whether a female larva develops into a queen or worker is the amount and type of food it receives. Too many queens could cause the colony to run inefficiently or break down, so to prevent this, some species raise their larvae in space-restricted cells where the food supply is controlled by the workers. This generally prevents the larvae from developing into queens. In species where this does not occur, excess queens are killed immediately after emerging.

A queen can actually mate with several males to produce offspring that may not be full siblings. A higher relatedness between workers, however, can reduce the incentive for such selfish acts. Focusing only on the genetics, an already-developed worker would benefit more from raising new sister workers than raising a new sister queen, as this could lead to the raising of nephews/nieces.

Conflict: matricide

There are three parties in favour of the queen’s death: laying workers, non-laying workers and, surprisingly, the queen herself. Going back to Hamilton’s rule, a model was developed by Bourke quantifying the cost–benefit ratio of the queen’s death. These are based on multiple variables including the number of offspring a laying worker can have between the queen’s hypothetical death and the death of the colony. The colony wouldn’t survive long without a queen, and raising new queens isn’t always successful due to the risky mating flight, where a virgin queen flies away from the safety of the nest to mate with males. Bourke’s model applies particularly well to annual colonies where eggs are laid together once a year, after which the colony members all die (apart from young mated queens).

After ordering the cost–benefit ratios of the queen’s death, it was found that laying workers have the most to gain, followed by non-laying workers, and then the queen herself. This is intuitive as, without the queen, laying workers can raise their own offspring. As we have seen in Fig. 1, this shouldn’t happen in the case where the queen has only mated once. However, if she has mated multiple times, the new workers may not be full sisters with the previous workers, and the relatedness would be less than $0.75$. Additionally, if the queen is old and laying fewer eggs (or unfertilised eggs due to food shortages), the benefit of a worker laying their own eggs increases. Once laying workers start attacking the queen, it creates a positive feedback loop: when the queen is injured, she will produce fewer eggs and the critical ratio for the non-laying workers can be surpassed, meaning that they start attacking the queen too. When the queen is seriously injured and no longer able to lay many eggs at all, she would gain a higher inclusive fitness from allowing the workers to lay eggs, since they would all be her grandsons. Thus, when the queen’s own critical ratio is met, she allows herself to be killed by her workers.

Female naked mole rat (Jedimentat44, CC BY 2.0)

Surprisingly, eusociality is also observed in mammals: specifically, two species of mole rat. The naked mole rat is an odd species, famous for its hairless appearance and high resistance to cancer. In captivity, they can live up to 31 years — an astonishing amount of time for a rodent! They live in underground nests under the rule of an aggressive queen who releases pheromones to discourage the workers from reproducing. Unlike the insects, the queen naked mole rat wasn’t born into her position, she had to fight for it! Her reign is also unstable: she will have to defend her crown from female workers. Mole rats (along with almost all mammals) are diploids, so at first glance, it seems that they are just as related to their offspring as to their siblings. However, naked mole rats are infamous for high rates of inbreeding, causing a high level of relatedness between the workers. A few of the males will have the role of mating with the queen, while the others function as workers. It still isn’t certain whether a high level of relatedness is required for eusociality to evolve, or whether it is a consequence of it, but clearly the two are linked.

Does altruism, the act of helping others at a cost to yourself, truly exist in the animal world? In the case of eusocial insects, sacrificing your own fertility to raise the queen’s eggs seems like a noble gesture, but we have seen that the workers only do this in order to pass on more of their genes. Once this benefit decreases, conflict occurs and the workers will continue to pursue their own optimal strategy — killing the queen to lay their own eggs. Can any of this theory be applied to humans? The majority of interactions occur between ‘unrelated’ individuals. For social dynamics, game theory would explain altruistic acts in terms of good karma, expecting others to return the favour later on (which is risky). Perhaps having this expectation can be selfish; however, a common assumption in game theory is that players act ‘rationally’ in the sense that they always try to maximise their payoff. I’d like to believe, though, that people don’t actually think like this and that acts of uncalculated kindness in humans occur more often than the theory would suggest.

[Banner image: Todd Huffman, CC BY 2.0;]

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Variations on Fermat: an agony in four fits

Fermat’s Last Theorem has been a source of fascination and the motivation for an enormous amount of mathematics over the last few centuries, both in attempts (eventually successful) to prove it and as the inspiration for other related questions.

This is the story of how an algebraic question inspired by Fermat’s Last Theorem morphed into an analytic question, which subsequently turned out to be expressed best as a geometric question, which could be answered using basic methods of plane geometry.

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Slide rules: the early calculators

Believe it or not, I never leave home without my trusty slide rule in my pocket. In the years before 1970, this would have been totally normal for any engineer, with the slide rule being the archetypal symbol of the engineering profession, much like how the stethoscope remains that of the medical profession. However, with the advent of the pocket calculator, the slide rule has completely vanished from public view and its demise is a wonderful example of a paradigm shift, as described by Thomas S Kuhn in The Structure of Scientific Revolutions.

I often get asked “What are you doing with that thing?” when I grab my slide rule to convert miles to kilometres or, once I’ve finished refuelling, to calculate my fuel economy. Most of my students have never seen a slide rule and are at first quite incredulous when I show them my Faber–Castell 62/82N. Not that I’ve ever managed to convince one of them to switch to a slide rule, but at least I normally manage to instil some interest in them for these mathematical instruments.

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The simplest difficult task

Most of us have heard of the RSA algorithm and how it’s very useful for cryptography. In order to crack it we need to be able to factor large numbers, but experience has told us that the problem of factorisation, while very simple to describe, is very difficult to do in practice. Yet there exists a problem that, though it might sound even simpler, is just as difficult. Continue reading

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Origami tesseract

Like Matt Parker’s drinking straws held together by pipe cleaners, this is an easy way to explore the symmetry of the
four-dimensional companion of the cube without fancy modelling software or a rapid prototyping machine. The wireframe oblique projection uses 24 sheets of A4 paper to create two classic origami cubes placed in parallel and joins their vertices with slanted beams. In a real tesseract these connections are made through a fourth axis but unfortunately I didn’t have the time to rip a hole into another dimension. Nevertheless, the model is a simple way in which to begin to understand the shape as a continuation of the dimensions we already experience. A point is to a line as a line is to a square, a square is to a cube as a cube is to a cubic prism. Continue reading

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Florence Nightingale, statistician

In her copy of Thomas à Kempis’ fifteenth century The Imitation of Christ is the inscription, in her own writing, “I only wish to be forgotten”. Although remaining a very private figure throughout her lifetime, this was not a desire that could ever be fulfilled. Since her death on 13 August 1910, at the age of 90, Florence Nightingale’s fame and the legend of the lady with the lamp has failed to dim. That her name has become a synonym for nursing is, however, misleading; and the public perception of her as a nurse does a great disservice to her life as a pioneering statistician.
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On the cover: dragon curves

Take a long strip of paper. Fold it in half in the same direction a few times. Unfold it and look at the shape the edge of the paper makes. If you folded the paper $n$ times, then the edge will make an order $n$ dragon curve, so called because it faintly resembles a dragon. Each of the curves shown on the cover of issue 05 of Chalkdust, and in the header box above, is an order 10 dragon curve.


Left: Folding a strip of paper in half four times leads to an order four dragon curve (after rounding the corners). Right: A level 10 dragon curve resembling a dragon


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