write an annotated bibliography for 4 sources.

Mathematics of the Human-Vampire Conflict Author(syf ' L Q R 6 H M G L Q R Y L ć Source: Math Horizons, Vol. 16, No. 2 (Nov. 2008yf S S 5 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/25678784 Accessed: 11-07-2017 01:08 UTC REFERENCES Linked references are available on JSTOR for this article: http://www.jstor.org/stable/25678784?seq=1&cid=pdf-reference#references_tab_contents You may need to log in to JSTOR to access the linked references. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://about.jstor.org/terms Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to Math Horizons This content downloaded from 74.137.237.188 on Tue, 11 Jul 2017 01:08:19 UTC All use subject to http://about.jstor.org/terms "Human-vampire conflict is special, since the predator actually turns the prey into his own kind, thus increasing his own population. Needless to say, this was not encountered in any other biological system!" Mathematics of the Human-Vampire Conflict Dino Sejdinovicf University of Bristol Do vampires exist? I'll bet most of you would say "Of course not!," but how can you really be sure? In a recent article, "Ghosts, Vampires and Zombies: Cine ma Fiction vs. Physics Reality," physicists C. Efthimiou and S. Gandhi attempt to offer a mathematical proof that vampires and other creatures can't possibly exist. But as we shall see, their argument has a hole in it, large enough that you can drive a stake through it! Here, we will deal mainly with the problem of the existence of vampires being creatures as presented in movies such as Vampires by John Carpenter or in the popular Blade trilogy. Efthimiou and Gandhi claim that the existence of such vam pires would lead to the inevitable extinction of humans in next to no time. According to myth, vampires feed on human blood and after sucking from a non-vampire victim, the victim itself becomes a member of the vampire race for the rest of its life (or death, or living death, since it is known that the vampires are the members of a larger class of night creatures called liv ing deadyf 7 R V X P X S R Q H P H D O I R U R Q H Y D P S L U H P H D Q V W K D t one human is gone and there is one more vampire in the world. This way, in a discrete time (e.g., having monthly updates on demographicsyf W K H V L P X O W D Q H R X V O L I H R U G H D W K R U \f of vam pires and non-vampires could be modeled by a set of differ ence equations: h{t+\yf K W \f-cv(tyf v(t+lyf Y W \f + cv(tyf . Here, v(tyf D Q G K W \f are the numbers of vampires and humans in the month t respectively, while c is the average monthly bloodsucking frequency (i.e., the average number of non-vam pires one vampire consumes in one monthyf . For the set of equations to be properly defined, starting con ditions h(0yf D Q G Y \f should be known. Efthimiou and Gandhi used quite advantageous assumptions for humanity: that in January 1600 (the authors present this date as the rough date of the first serious appearance of these dark creatures in folkloreyf there existed only one vampire against 500 million non-vam pires, which was the world population at the time. Another assumption is that bloodsucking occurs only once per month, which is a very mild and cautious assumption since in the movies, you can see that vampires eat all the time. According to the given model, this would mean that c = 1, and therefore v(t +1yf Y L \f. Thus, since v(0yf Z H J H W D V L P S O H H T X D W L R n v(0=2', so the vampire population grows exponentially! Consequently, since v(29yf L V J U H D W H U W K D Q P L O O L R Q , we conclude that the human race would cease to exist some two and a half years after the appearance of the first vampire, approximately thirty years before Rene Descartes would get to think that he is since he thinks. Needless to say, vampires would also die out soon afterwards, because of the absence of human blood. It seemed like Efthimiou and Gandhi gave undeniable proof and closed the ever exciting matter of vampire existence. However, the defiance of common folks was great. On the web site digg.com, one could read skeptical comments stating that the authors did not take into account that the extinction of the human race would not be allowed by Blade, Buffy and Chuck Norris; that garlic necklaces were quite popular in the seventeenth century; and that natural laws could not be applied to supernatural beings. On second thought, the model has not accounted for the birth-rate of non-vampires and death-rate of vampires (actual ly the death-death-rate since they are already dead, but when they die again they should stay dead but stop being livingyf G X e to close encounters with stakes, garlic and holy water. Also, vampires are presented exclusively as greedy consumers: a rational strategy of managing their human resources is not con sidered. In fact, such a rational policy was proposed over twenty years ago by a team of Austrian mathematicians, led by R. Haiti and A. Mehlmann. They used mathematical optimization and control theory to model a dynamic confrontation between the human race and vampires taking into account optimal bloodsucking strategies of the vampire population. The authors were irritated that this matter was only being consid ered by anthropologists and producers of low-budget movies. They presented a more realistic model of the conflict: v = - av + cv h' = nh- cv. Here, v and K are the rates of change of vampire and human population in time, a is the death-rate of vampires 14 NOVEMBER 2008 This content downloaded from 74.137.237.188 on Tue, 11 Jul 2017 01:08:19 UTC All use subject to http://about.jstor.org/terms MATH HORIZONS caused by stakes and other good stuff, while n is an increase in human population by common reproduction. As before, c is the bloodsucking frequency. This is an example of the more general Lotka-Volterra set of equations, common for modeling predator-prey interactions in biological systems. Human-vam pire conflict is special, since the predator actually turns the prey into his own kind, thus increasing his own population. Needless to say, this was not encountered in any other biolog ical system! Hard and Mehlmann put themselves in the position of vam pires and tried to solve what was to be adequately named the Transylvanian problem: determine the optimal bloodsucking frequency c(tyf Z L W K U H V S H F W W R W L P H Z K L F K Z R X O G P D [ L P L ] H W K e objective function representing total utility per vampire, assuming vampires have a utility function U(cyf 7 K H X W L O L W y U(cyf L V G H U L Y H G E \ W K H D Y H U D J H Y D P S L U H I U R P E O R R G F R Q V X P p tion at rate c, while total utility per vampire is what determines joy and happiness inside the vampire community. This prob lem was solved for three different kinds of vampires: (ayf D V \ P S W R W L F D O O \ V D W L D W H G Y D P S L U H V Z K H Q 8 ; (byf E O R R G P D [ L P L ] L Q J Y D P S L U H V Z K H Q 8 ; (cyf X Q V D W L D E O H Y D P S L U H V Z K H Q 8 ! , which, it should be noted, is also a large step forward in the chaotic classification of vampires. Clearly, Efthimiou and Ghandi have underestimated the mathematical capabilities of the vampire community. Strikingly, there are strategies for all three kinds of vampires which would approach a long-run equilibrium for the value of the stock of humans per vampire x{tyf K W \flv(tyf , Q R W K H U Z R U G V Y D P S L U H V F R X O G K D Y H D S O D Q W o manage their human resources in order to keep both human race?and thus, their own?in existence. Later, these same sci entists used Hopf bifurcation to strengthen their results and succeeded in modelling the human-vampire conflict so that optimal bloodsucking rates change periodically, which is more concurrent with archived empirical evidence in popular litera ture. These works created swift reaction by the counter-vampire scientific community. Political economist, D. Snower expressed his indignation with the efforts of Hard and others to help the vampires strengthen and consolidate their bloodsuck ing strategies, and came forth with a macroeconomic optimal strategy of the destruction of vampires. His philanthropic plan consists of a smart allocation of labor services producing eco nomic goods onto the production of useful counter-vampire instruments, such as stakes, garlic, or rosary beads. However, by standard application of the maximum principle, Snower reached the surprising result that a complete destruction of vampires would not be socially optimal for humans either! From the research done by Hard, Mehlmann and Snower, it is clear that the relations between vampires and non-vampires are much more sophisticated than those presented by Efthimiou and Ghandi. Assuming that both vampires and non vampires are intelligent beings, choosing their actions ration ally, science actually sends a rather pacifist message that the optimal plan for both races is life (or death...yf W R J H W K H U $ Q d that all should live (dead or notyf K D S S L O \ H Y H U D I W H U . Except for the unfortunate fellows who get their blood sucked dry. B Further Reading C. J. Efthimiou and S. Gandhi, Ghosts, Vampires and Zombies: Cinema Fiction vs Physics Reality, arxiv-.physics1060859, 2006. R. F. Haiti and A. Mehlmann, The Transylvannian Problem of Renewable Resources, Revue Francaise d'Automatique, Informatique et de Recherche Operationelle, Vol. 16 (1982yf , 379-390. R. F. Hartl, A. Mehlmann, A. Novak, Cycles of Fear: Periodic Bloodsucking Rates for Vampires, Journal of Optimization Theory and Applications, Vol. 75 (1992yf . S. King, Salem's Lot, New American Library, New York, New York, 1969. D. Snower, Macroeconomic Policy and the Optimal Destruc tion of Vampires, Journal of Political Economy, Vol. 90 (1982yf . Research topic: A three-week summer program for Arithmetic of L-functions graduate students undergraduate students mathematics researchers Education Theme: undergraduate faculty Making Mathematical secondary school teachers Connections math education researchers IAS/Park City Mathematics Institute (PCMIyf June 28-July 18, 2009 Park City, Utah Organizers: Cristian Popescu, University of California, San Diego; Karl Rubin, University of California, Irvine; and Alice Silverberg, University of California, Irvine. Graduate Summer School Lecturers: David Burns, Kings College, London; Henri Darmon, McGill University; Benedict Gross, Harvard University; Guido Kings, Regensburg University; Manfred Kolster, McMaster University; Cristian Popescu, University of California, San Diego; David Rohrlich, Boston University; Karl Rubin, University of California, Irvine; John Tate, University of Texas at Austin; and Doug Ulmer, University of Arizona. Clay Senior Scholars in Residence: Benedict Gross, Harvard University; and John Tate, University of Texas at Austin. Other Organizers: Undergraduate Summer School and Undergraduate Faculty Program: Aaron Bertram, University of Utah; and Andrew Bernoff, Harvey Mudd College. Secondary School Teachers Program: Gail Burrill, Michigan State University; Carol Hattan, Vancouver, WA; and James King, University of Washington. Applications: pcmi.ias.edu Deadline: January 28, 2009 IAS/Park City Mathematics Institute Institute for Advanced Study, Princeton, NJ 08540 Financial Support Available WWW.MAA.ORG/MATHHORIZONS 15 This content downloaded from 74.137.237.188 on Tue, 11 Jul 2017 01:08:19 UTC All use subject to http://about.jstor.org/terms