on vampires and stochastic processes


Welcome vampire enthusiasts! See the response posted on March 31.

The movie Twilight came out on DVD came out earlier in the week. This movie about teenage vampires made a lot of money at the box office, and I have to admit that I’m a little curious to see what all the fuss is about. But I can’t get into the whole vampire thing. I have a great deal of skepticism about vampires.

Here’s my problem with vampires. I have a hard time believing that there would be just a few vampires out there and that the existence of vampires would be such a well-kept secret. After all, they reproduce rather easily (a single vampire could create thousands of offspring, whereas there are limits to human reproduction) and vampires don’t die easily. If there were vampires, they would almost certainly outnumber humans (but then vampires would run out of food).

This argument becomes even more overwhelming if you model a vampire population as a branching process or birth-death process and assume that each vampire in the population has probability Pj of producing j offspring (with j=0,1,2,… ). The vampire population would either explode or die out, depending on the expected number of offspring per vampire. But if you take into account the fact that vampires live many, many generations (they’re virtually immortal) and may create thousands of offspring, the population explodes (if you assume that each vampire creates at least one vampire, on average, before it dies). With those numbers, vampires would not be living under the radar–they would be everywhere!

I have yet to see a vampire movie that implicitly assumes that there is a reasonable model for vampire population dynamics (using a stochastic process framework or something else). And frankly, I’m pretty disappointed. Until I am offered a reasonable explanation for why there aren’t more vampires, I won’t be able to jump on the vampire bandwagon. If I had free time, maybe I would write a mathematically consistent vampire novel.

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47 responses to “on vampires and stochastic processes

  • Kevin W

    So the vampires have a tragedy of the commons problem: each one wants to reproduce, but if they all reproduce at maximum speed the entire community gets ruined.

    Maybe they deal with this the same way we do, with rules. There’s a theme in vampire fiction of a cautionary tale about a group of vampires that reproduced quickly, attracted a lot of attention which turned into an inquisition, and got wiped out. That leads to a code of vampire law that, among other things, puts limits on reproduction.

    Maybe the branching factor is lower than you think. Vampire reproduction seems a lot like mafia reproduction, and the mafia population never exploded. On “The Sopranos,” Tony had a real hard time finding candidate gangsters that were strong enough to be useful, weak enough that they don’t become rivals, and agreeable enough to be tolerable indefinitely. A wise vampire would be particular about who gets turned into an immortal monster.

    Maybe vampires have some sort of biological adaption that limits their growth, similar to real-world pinnacle predators.

    But really the objective of a vampire movie isn’t to be realistic, the objective is to be entertaining. We could debunk warp drives, lightsabers, Frodo’s One Ring, or Santa Claus, too. The whole point is to suspend disbelief a little bit and have fun. :)

  • Jim Orlin

    It’s a relief to have a mathematical argument that vampires don’t exist. Here is yet another reason why Operations Research lets you sleep better at night :)

  • Marco

    Hi Laura, there are a lot of mathematically (:P) consistent vampire novels. Some examples include Ann Rice’s books and a couple of role playing games by Whitewolf.
    They found several “reasonable” explanations:

    1. vampires do not reproduce easily. In order to create a new vampire a long ritual must be followed and the candidate vampire must accept to become a vampire (under the curtains there is the christian interpretation of “free will”). Moreover, the vampire society has very strict rules about reproduction. A vampire is usually not allowed to create new vampires, he/she could be punished with his/her own death.
    2. vampires die easily… killed by other vampires.

    Hope this could be of help for a math model of vampire populations :)
    cheers,
    Marco

    ps: I do not believe in vampires ;)

  • Laura

    Thanks for all the vampire pointers. I suppose I should lighten up, add Twilight to my queue, and enjoy :)

  • Paper Hand

    Well, there are Slayers and the like keeping the population in line …

    I know I saw somewhere, I forget where, a calculation of the vampire population of Sunnydale, assuming that Buffy slays them at roughly the same rate they’re produced. I’ve long since lost it, though.

  • Gil-Galadh

    Isn’t it obvious – the werewolfs kill them :)

  • Rossi

    It’s all explained in the film that started this whole line of enquiry. To create a new recruit, a vampire must bite a human, but not kill them. This is incredibly difficult, as vampires have voracious appetites and kill in a kind of berserker frenzy. It follows that only the most disciplined, and arguably, intelligent and charismatic vampires are able to “reproduce”.
    Also, vampires would have evolved over time with humans and so developed some kind of negative feedback to keep themselves in check, especially given the delicate nature of their covert existences.

  • Laura

    Wow, you guys know a lot about vampires! Thanks for the feedback. I guess a vampire population could be at steady-state given these assumptions, but I still have some doubts. I am willing to believe that most–but not all–vampires would behave in a responsible and rational way. What if just a small fraction of vampires (say, 1 in 100) goes rogue and creates like another 1000 vampires? The vampire population would slowly and exponentially grow.

    When you take into account the fact that vampires probably won’t ever go extinct because the last vampire can essentially live forever (there is not another vampire to kill it), then maybe I am over-analyzing this. But then a werewolf could kill the last vampire, which reinforces my original point. I confess that I know less about werewolves than vampires, so I’ll have to save that thought for another day :-)

    Thanks again for reading!

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  • joe

    Check out the novel “Wasting The Dawn.” Not only is it one of the better vampire novels to come out recently, (Published by IDW, the same people who brought you the 40 days of night comic book.) part of the plot does have to do with vampire population dynamics.

  • Steve

    “What if just a small fraction of vampires (say, 1 in 100) goes rogue and creates like another 1000 vampires? The vampire population would slowly and exponentially grow.”

    That reminds me of the Fermi Paradox – most arguments about why we don’t see any evidence of extraterrestrial life aren’t conclusive, and all it would take is for one civilisation to avoid the filter to conquer the galaxy.

    You’ve discovered the vampire Fermi Paradox!

  • JJason

    Have you read “Vampire Population Ecology”? You can find it summarized on everything2 – it finds a stable vampire population given the presence of humans who can kill off vampires, specifically Buffy the Vampire Slayer.

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  • MissElizaBennet

    After discovering this link through Boing Boing and discussing with my 14 year old daughter, who is a Buffy, Angel and Twilight fan (among many), she became really agitated. Much has been made of which vampire myth you rely on re how vampires are created, and, if you refer to Buffy series 7, you will actually discover a vampire army! Thanks for providing such an opportunity for animated and passionate discussion.

  • Geoff

    It has, in fact, been done. In his last novel, Blindsight, Peter Watts casts ‘vampires’ as a separate hominid subspecies from prehistory who preyed on homo sapiens, but reproduced the old fashioned way amongst themselves instead of turning humans into vampires. They even had a mechanism for hibernating for decades at a time while prey population stocks recovered.

    That’s actually probably the least weird thing in the novel, which is about as odd a first contact story as you’re likely to find.

  • Marjorie

    The Buffy vampire model to which Paper Hand refers is probably Vampire Population Ecology by Brian Thomas: http://filk.buffistas.org/miscellaneous/vamp_ecology.pdf.

  • Notger

    Laura, you might want to check Viktor Pelewins “Fifth Empire”, a very strange modern vampire novel that is mathematically consistent.

    In it, human vampires are merely the vessels for immortal vampire “souls”, or more precise, for their tongue, an additional organ that – after occupation – resides on the back of your mouth and in your brain and controls you. This “vampire tongue” moves from human shell to shell, being physically mortal but not able to die from old age, refreshing itself from every human it was transported in.

    In this novel, vampires do not breed. The number of tongues that exist is limited from ancient times on and can only degrade. Thus the need for secrecy … if anyone would find out, they community would be further weakened.
    (Well, to be honest, Pelewin does not go into details concerning the breeding … maybe there is some way to create new tongues, but it was never mentioned.)

    I like the idea of abandoning that romantic anthropocentric view of the “elite-humans” that vampires are often depicted. Humans are vampire cattle.

    In fact, there is a lot more in this book (learned a lot about Russia, modern Russia, media and cargo cult). You should definitely check it out.

  • Tordr

    First off if you have one vampire that goes rogue and creates lots of other vampires, then that vampire will have to do that in some remote location to stay undetected. He or she will feed off the population until all the people are either dead or vampires. This process will be a runaway process and has the potential to destroy the world, but at some point the threat will be discovered and once the threat is identified both vampires and humans will try to defeat it.

    Using the same argument you could also ask why there are cows? Humans eat cows at a much greater speed than they can reproduce. And if everyone in the world tried to eat one cow during the next month there would not be a single cow left. But cows still survive because we let them survive. Although birth-death models would predict that cows where wiped out tomorrow if we use ordinary birth-death models for predators.

    Another interesting fact is if there exists vampires, then why are we not slaves to the vampires? If all vampires live longer, are smarter and more superior to humans it would make sense that they where the masters and we just slaves. See Stargate for the relationship between Goa’uld and humans. The humans are just hosts and like cattle for the Goa’uld. In the same way it would make more sense for the vampires to be open about it and dominate humanity instead of hiding in the shadows.

    Humans can rebel against vampires, but it only takes one vampire to spawn thousands, so over time they would naturally have become the dominant force.

  • Nicco

    What about the revengeful mob that is created every time a human is made into a vampire? For each person transformed many relatives/friends etc probably will go hunt the original vampire which is kind of a density dependent effect. So the more new vampires the more new vampire hunters, clearly a strong regulating effect (plus the hunters only need to find an “early” vampire to destroy a whole bunch, right?).

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  • James

    I always thought that to turn into a vampire you had to not only be bitten, but also had to drink the blood of the vampire as well (see http://eruditebaboon.livejournal.com/13294.html). It’s a bit more complex than just running around biting as many people as you can and turning them all into insta-vampires. I – I think that’s zombies, right?

  • Nerdcore Steve

    Anne Rice seemed to answer this question by saying that vampires typically don’t have the fortitude for immortality. Within less than a hundred years they commit suicide and let themselves die because they can’t stand being in a world that is so different from the one they started in, or they kill themselves out of loneliness.

  • Josh

    Try “I Am Legend” (the book, not the movie), which addresses this point by having the entire population of the planet overrun by vampires, to the point where only one human remains.

  • Ryan

    Discussing this using the particulars of you’re favorite vampire fiction is a bit pointless. Every writer, for the sake of a better story, comes up with his own little twists and variations. Complicating this they usually do this based entirely or in part on what other writers and general pop culture have added to the pot.

    But if we’re going to talk about the what if of vampires existing in the really real world then its important to look at claims of vampires existing in the really real world.

    http://en.wikipedia.org/wiki/Vampire

    In folklore vampires are pretty varied. But the modern pop culture vampire is based largely off eastern European legend of the middle ages through the 18th century. Here vampire creation is extremely random and limited. Perhaps a black cat jumps of the body, or the person was a witch or otherwise involved with demons. Or maybe they just didn’t get baptized, or receive last rights.

    Long story short the vampire is some one who is cursed/damned during or just after life, and thus comes back as a monster. The concept of vampires begetting vampires (“breeding”) isn’t very common. Occasionally it is mentioned though mostly in more recent times. We see a bunch of people die from slow wasting sickness a vampire is blamed, and steps are taken to keep the victims from becoming vampires themselves.

    This sort of vampire is more of a mindless killing machine, with very limited agency. There is no deliberate planning, or hierarchical society. In fact there is usually only one vampire in any given place, with any others identified afterwords by chance. These vampires can occasionally mesmerize people to lure them in, or make them do the vampires bidding. But the mesmerized are not, at all, vampires.

    These vampires can not relocate, they are bound to the grave in which they were originally interred. Even if discovered, or they run out of food they always return to the same spot every morning.

    So the vampire population in this estimation will always be very limited, a few individuals here and there easily identified and destroyed. Even in cases where the condition spreads it does so very slowly and is tied to a very specific area. But since the the vampire crop up through such varied and seemingly random ways there will all ways be some.

    Even in later demonology and Kabala where vampires are the descendants of Cane or Lilith they would be severely limited in number. Cane’s kids are marked and we’re all supposed to stay away from them, so reproduction options are far more limited than you’re average human (and they do reproduce in the normal human way). So his offspring exist as an isolated population, they have to wander so they’d be spread even thinner (think if the population of Fiji or some such were spread out world wide). And since they have (biblical) human life spans they wont be out breeding us through pure longevity. Lilith vampires were all created through sexual reproduction in one giant batch 1000′s of years ago. When they reproduce sexually they usually don’t spawn other vampires. Their life spans are varied and not always unlimited. Neither of these types spread their condition to victims.

    Even what are usually referred to as vampires from other cultures, like Gaelic kelpies and Japanese Kappas, are usually an angry ass nature spirit. Tied to a specific place or kind of place. They don’t reproduce per say just sort of pop when whatever they’re associated with does. They don’t spread the vampirism either.

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  • Orval

    It’s pretty simple Darwinism, really. Either the predator/pathogen will be so successful as to kill all its prey (and then die of starvation), or the prey will come up with a comeback (like Buffy), either wiping out the dependent pathogens, or at worst finding some kind of balance, a la Zebras and Lions if you will.

    But me, I kind of think of vampires as illegal immigrants; trying desperately Not To Be Noticed. Rather than being our New Overlords (which I for one, of course, welcome) maybe they are mostly like illegally here hispanic maids that you don’t have to pay social security for.

    Because it strikes me that being a vampire is a pretty dangerous occupation: all the literature (outside of the ridiculous sparkly set) seems to agree that vampires don’t care much about their offspring: new vampires have a survival rate right up there with smelt and plankton. And if you get too busy, you find the peasantry getting all active with the garlic and torches and pikes and beheadings thing.

    So maybe there is a natural feedback loop here. Which implies being scared of and taking precautions, might not be all that bad an idea. Just sayin’.

  • gttim

    Vampires seem to die very easily. There are a bunch of ways to kill them, and even high school cheerleaders seem to be able to do so. If they all died like Paul Reubens, the world would be a much funner place.

  • Vnend

    Obviously, given your math, we are all vampires, and have been for generations. Vampire stories are simply our subconscious racial guilt working its way out. The fact that we killed off all our hosts forced us to ingest ‘food’, but that is not really enough to sustain us, so now we all age and die. Except the Wise Ones who kept a small but sufficient breeding pool of our original hosts alive and secluded…

    Yes, if you look at the math like a standard fox-and-hare or other predator/prey situation with no feedback in the system then you get a world overrun by vampires (and the only one left is the meanest, strongest one of the bunch.) But even if they only maintain the intelligence they had when human, the vampires would ‘quickly’ come to realize that they cannot all keep making new vampires all the time.

    At an SF con in Dallas about 25 years ago, I was part of a group chatting about writing and ideas. Two of the group were George R.R. Martin and Roger Zelazny. I pointed out that, in most cases, the ‘idea’ that (some folks) worry about ‘protecting’ is so broad that, even if someone else learned it, the story they would come up with is completely different from the one the original person was thinking. I trotted out the example idea ‘consider the ecology of vampires’. Roger later published the short story ‘Dayblood’, which, as expected, took the idea a completely different direction than I was thinking. I don’t know if any of George’s stories included that meme, but I suspect that it was there, in the background. Stephen Brusts’ ‘Agyar’, published in the 90′s, certainly seemed to be aware of the question. But then, I wasn’t the first person to think of it either.

    My long-winded point is, if you have evidence of a stable system that has existed for many generations, then there are some factors helping to keep it in balance. Vampires, given the folklore, are part of such a system. It doesn’t really matter if the mechanism is social, biological or mystical. Finding out what it is might be nice, but not required. The downside is when someone writes a story and changes the definition of ‘vampire’ to the point that all the ‘standard’ checks and balances fall apart and you find yourself thinking ‘that was stupid’ instead of being able to enjoy the story.

  • LSK

    Simple.

    Not all people who are bitten by vampires become vampires.

  • Geoff T

    Simple.

    Twilight’s vampires aren’t really vampires. Real vampires don’t glitter in sunlight. Real vampires explode. ;-)

  • Christina

    I like your points.

    If you take the time to read all the books from the Twilight series you will get all the answers to your questions.

    In the story there is more then ‘just a few’ of them. There is about as many of them, if not more, as there are us. They do not create others of them selves anymore (or are outlawed to) because they know of the food problem.

    It is just like with us. There are…what….9 billion? of us? We are going to keep growing in number and our food will die off and we will kill the land so we may not grow our own…. we are just as fucked as you say they would be – just we are doing it slower.

  • ChrisB

    It’s not just vampires, it’s also zombies; they also increase exponentially.
    Come to think of it, zombie movies are surely inaccurate in their portrayal of flesh-eating mobs. When the graveyards gape and give forth their dead there must, actuarially, be a higher proportion of the very old than is usually rendered. At least in the early days of the epidemic, there must be a near-majority of octogenarians, which you would think would be easier to deal with.

    Other unanswered questions are
    * what is the signal that inhibits zombies from eating other zombies?
    * Could we synthesize it, thus enabling us to walk unhindered through the mob?
    * Why doesn’t the zombie effect affect animals? Except police dogs?
    * given that what zombies actually want is brains, how do they get them without tools? I couldn’t bite a skull open, and I’m not sure anybody could. It would be like trying to bite your way into a coconut (and have you ever tried to put two canine teeth marks on a person’s neck? You have to practically dislocate your jaw).

  • Maidhc

    I once developed a computer model based on the assumption put forth by Nicco at 19, i.e., that the probability of humans becoming vampire hunters is a function of the number of humans killed by vampires. An equilibrium is possible.

  • Ken Kills

    I would think the best three possible models would be:

    1) The probability of “conception”, as it were, is ridiculously small. If the vampire needs to feed once a week, and the probability of conception per feed is 1 in 10,000, you’d get an average of 1 new vampire per 192 years. If you factor in natural wasteage (we know they can be killed), then the population should be fairly static / grow very slowly.

    2) If you go for a “viral egg” model of reproduction, i.e. the vampire has 10 units of “vampire juice”, it takes one unit of “vampire juice” to create a new vampire, but each use of “vampire juice” makes the vampire less powerful, then you could end up with a small population over time, particularly if they’re more likely to die the weaker they are.

    3) The easy explanation is that they can reproduce once, and then have a very very very slim chance of reproducing a second time. You’d then get a very gradual increase in population (maybe one or two per generation of vampires), and then you’ve got the natural wasteage on top

  • Clarissa

    I see a lot of other folks have made very good points already. One other way of dealing with the issue is used by Tanya Huff in her “Blood” novels: that vampires are naturally *incredibly* territorial about hunting grounds, to the extent that if more than one vampire is present in the same city (though some larger cities might be adequately dividable, I don’t remember what the specific limits are) they’ll end up killing each other.

    This not only controls the population of vampires in a given area, therefore avoiding over-hunting of humans, but is a somewhat inhibiting factor in creating more vampires. Most vampires would be driven to make their close loved-ones immortal, but the resulting inability to remain together makes it a less attractive idea.

  • objection!

    1) Not everyone who gets bit by a vampire turns into one, most of them die.

    2) Actual vampire slayers keep the population down, such as Jefferson Twilight, the Blacula killer from The Venture Bros.

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  • Rich

    Why don’t Vampires rule the settings they inhabit? This question is answered differently or not at all in most settings.

    Some things that are sometimes true:
    1. Sunlight severely burns them or kills them.
    a. Yet vampires never seem to be able to figure out how to cover up in protective gear.
    b. Sometimes ultra violate light from a source other than the sun will work for hunters.

    2. Crosses, holy-ground and holy water keep them at bay (sometimes it is the religious belief of the vampire, the humans [or sentient thought in general] or a supposed true faith).
    a. Vampire hunters are safe at night from close combat attacks directly from vampires.
    b. A missile launcher, grenade, pipe bomb, mercenaries, inferred sensors and a high powered firearm could bypass this restriction.

    3. If all the holy stuff affects vampires, then the church(s) knows about vampires.
    a. Vampires are being hunted by human organizations such as the Catholic Vatican who has over a thousand years of experience, vast holy ground holdings and a great number of priests capable making holy water and consecrating holy ground (I would imagine worship wound need to take place at a given site for holy ground to keep its sting).
    b. Vampires do not want to have a swat team of holy warriors with super soakers, paintball guns, holy water tipped needle guns, shotguns with blessed rosary beads, ultraviolet floodlights and a dozen priests consecrating there home as holy. Not to mention explosives, conventional firearms and the litany of death humanity can reign down. A sponge soaked with holy water is deadly. Fire trucks filled with holy water.

    4. Vampire can not cross running water. Perhaps it has holy water in it or some such.
    a. I would think that there would be a range so that a vampire on an airplane can cross running water. Perhaps on a ship, in a crate whimpering.
    b. Vampire hunters would know about this and use it tactically.

    5. Vampires cannot abide garlic.
    a. Looking back at #3 we can add all sorts of weapons with garlic. Garlic spray, garlic shotgun shells. Heck a small fleet of trucks can dump tons of garlic around a vampires house, or just spray concentrated garlic all over. Heck we could fumigate areas with liquid garlic like we are spraying for mosquitoes or such.

    6. Vampires can be burned to death.
    a. Rubbing two sticks together and setting there house on fire will do it.
    b. The house-fire down the street was arson. Some vampire was hiding in the basement.

    7. Vampire hunters could have armor.
    a. I bet it would be hard to bite though Kevlar or get to flesh though a stormtrooper (Star Wars) style clamshell armor. Vampire hunters could have armor with little packing bubbles all over full of garlic or holy water, baptize themselves with holy water before going in to battle (with spongy water soaking clothing).
    b. Heck crosses all over could be armor.

    8. Vampires are screwed.
    a. Even if we get a super powerful vampire unapproachable by a Vatican hunter squad, we have bunker busters, gas leaks and all sorts of death from bombers and fighters.

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  • CaptainBlack

    The recent film daybreakers seems to refer to the problem of vampire over population (population explosion might be a suitable term given the effect used when a vampire expires).

    CB

  • Tasha

    Do you think that it was possible, that back in the 18 century, that could be the reason why people believed in folklore, such as zombies and vampires? That maybe because there was no way to disprove that they exist?

  • Horribly Cheerful

    It’s about time someone started paying attention to this whole VAMPIRES CANNOT LIVE IN SECRET thing. Unless they ran off to another world, let’s say, Faerie? Where all things magical and mythical exist.

    God love the fey.

    (Yes I do realize that none of this is real, but why not imagine it?) :)

  • ryan

    @44 I think your concept of “the fey” is a little skewed by fantasy fiction there is no place called faerie where these things reside. In old celtic mythology and the folklore that follows from it (irish, british, welsh, scottish,and some french) these things existed in the real world. Usually certain hills, woods, and ruins were regarded as the home of certain creatures or gods. Many of these places still exist and are avoided or treated with a certain respect by the locals. Even the words fey and faerie/fairy are recent anglicisations of the original terms. Collectively they were call sidhe (“she”) meaning hill or from the hills. Males were fae sidhe females ban sidhe meanin man from the hills and woman from the hills respectively. Fae sidhe became both fey and faerie/fairy later, and became a common term for the group as a whole. Ban sidhe (obviously) became banshea and became associated with one specific critter.

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