Mathematics of zombies

What do you do if there is a Zombie attack? Can mathematics help? This page is (humorously) dedicated to collecting links to papers or blog posted related to the mathematical models of Zombies.

George Romero’s 1968 Night of the Living Dead, now in the public domain, introduced reanimated ghouls, otherwise known as zombies, which craved live human flesh. Romero’s script was insired on Richard Matheson’s I Am Legend. In Romero’s version, the zombies could be killed by destroying the zombie’s brain. A dead human could, in some cases be “reanimated,” turning into a zombie. These conditions are modeled mathematically in several papers, given below.

1. When Zombies Attack! Mathematical Modelling of a Zombie Outbreak!, paper by Mathematicians at the University of Ottawa, Philip Munz, Ioan Hudea, Joe Imad and Robert J. Smith? (yes, his last name is spelled “Smith?”).
2. youtube video 28 Minutes Later – The Maths of Zombies , made by Dr James Grime (aka, “siningbanana”), which references the above paper.
3. Epidemics in the presence of social attraction and repulsion, Oct 2010 Zombie paper by Evelyn Sander and Chad M. Topaz.
4. Statistical Inference in a Zombie Outbreak Model, slides for a talk given by Des Higman, May 2010.
5. Mathematics kills zombies dead!, 08/17/2009 blog post by “Zombie Research Society Staff”.
6. The Mathematics of Zombies, August 18, 2009 blog post by Mike Elliot.
7. Love, War and Zombies – Systems of Differential Equations using Sage, April 2011 slides by David Joyner. Sage commands for Love, War and Zombies talk. This was given as a Project Mosaic/M-cast broadcast.
8. Public domain 1968 film Night of the Living Dead by George Romero.

Paley graphs in Sage

Let $q$ be a prime power such that $q\equiv 1 \pmod 4$. Note that this implies that the unique finite field of order $q$, $GF(q)$, has a square root of $-1$. Now let $V=GF(q)$ and

$E = \{(a,b)\in V\times V\ |\ a-b\in GF(q)^2\}.$
By hypothesis, $(a,b)\in E$ if and only if $(b,a)\in E$. By definition $G = (V, E)$ is the Paley graph of order $q$.

Paley was a brilliant mathematician who died tragically at the age of 26. Paley graphs are one of the many spin-offs of his work. The following facts are known about them.

1. The eigenvalues of Paley graphs are $\frac{q-1}{2}$ (with multiplicity $1$) and
$\frac{-1 \pm \sqrt{q}}{2}$ (both with multiplicity $\frac{q-1}{2}$).
2. It is known that a Paley graph is a Ramanujan graph.
3. It is known that the family of Paley graphs of prime order is a vertex expander graph family.
4. If $q=p^r$, where $p$ is prime, then $Aut(G)$ has order $rq(q-1)/2$.

Here is Sage code for the Paley graph (thanks to Chris Godsil, see [GB]):

def Paley(q):
K = GF(q)
return Graph([K, lambda i,j: i != j and (i-j).is_square()])



(Replace “K” by “$K.\langle a\rangle$” above; I was having trouble rendering it in html.) Below is an example.

sage: X = Paley(13)
sage: X.vertices()
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]
sage: X.is_vertex_transitive()
True
sage: X.degree_sequence()
[6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6]
sage: X.spectrum()
[6, 1.302775637731995?, 1.302775637731995?, 1.302775637731995?,
1.302775637731995?, 1.302775637731995?, 1.302775637731995?,
-2.302775637731995?, -2.302775637731995?, -2.302775637731995?,
-2.302775637731995?, -2.302775637731995?, -2.302775637731995?]
sage: G = X.automorphism_group()
sage: G.cardinality()
78



We see that this Paley graph is regular of degree $6$, it has only three distinct eigenvalues, and its automorphism group is order $13\cdot 12/2 = 78$.

Here is an animation of this Paley graph:

The frames in this animation were constructed one-at-a-time by deleting an edge and plotting the new graph.

Here is an animation of the Paley graph of order $17$:

The frames in this animation were constructed using a Python script:

X = Paley(17)
E = X.edges()
N = len(E)
EC = X.eulerian_circuit()
for i in range(N):
X.plot(layout="circular", graph_border=True, dpi=150).save(filename="paley-graph_"+str(int("1000")+int("%s"%i))+".png")
X.delete_edge(EC[i])
X.plot(layout="circular", graph_border=True, dpi=150).save(filename="paley-graph_"+str(int("1000")+int("%s"%N))+".png")


Instead of removing the frames “by hand” they are removed according to their occurrence in a Eulerian circuit of the graph.

Here is an animation of the Paley graph of order $29$:

[GB] Chris Godsil and Rob Beezer, Explorations in Algebraic Graph Theory with Sage, 2012, in preparation.

Sestinas and Sage

According to [1], a sestina is a highly structured poem consisting of six six-line stanzas followed by a tercet (called its envoy or tornada), for a total of thirty-nine lines. The same set of six words ends the lines of each of the six-line stanzas, but in a shuffled order each time. The shuffle used is very similar to the Mongean shuffle.

Define $f_n(k) = 2k$, if k <= n/2 and $f_n(k) = 2n+1-2k$, if $k > n/2.$ Let $p = (p_1,...,p_n) \in S_n$, where $p_j = f_n(p_{j-1})$ and $S_n$ is the symmetric group of order $n$. From [2], we have the following result.

Theorem: If p is an n-cycle then 2n+1 is a prime.

Call such a prime a “sestina prime”. Which primes are sestina primes?

Here is Python/Sage code for this permutation:


def sestina(n):
"""
Computes the element of the symmetric group S_n associated to the shuffle above.

EXAMPLES:
sage: sestina(4)
(1,2,4)
sage: sestina(6)
(1,2,4,5,3,6)
sage: sestina(8)
(1,2,4,8)(3,6,5,7)
sage: sestina(10)
(1,2,4,8,5,10)(3,6,9)
sage: sestina(12)
(1,2,4,8,9,7,11,3,6,12)(5,10)
sage: sestina(14)
(1,2,4,8,13,3,6,12,5,10,9,11,7,14)
sage: sestina(16)
(1,2,4,8,16)(3,6,12,9,15)(5,10,13,7,14)
sage: sestina(18)
(1,2,4,8,16,5,10,17,3,6,12,13,11,15,7,14,9,18)
sage: sestina(20) (1,2,4,8,16,9,18,5,10,20)(3,6,12,17,7,14,13,15,11,19)
sage: sestina(22) (1,2,4,8,16,13,19,7,14,17,11,22)(3,6,12,21)(5,10,20)(9,18)

"""
def fcn(k, n):
if k<=int(n/2):
return 2*k
else:
return 2*n+1-2*k
L = [fcn(k,n) for k in range(1,n+1)]
G = SymmetricGroup(n)
return G(L)



And here is an example due to Ezra Pound [3]:

                                  I

Damn it all! all this our South stinks peace.
You whoreson dog, Papiols, come! Let’s to music!
I have no life save when the swords clash.
But ah! when I see the standards gold, vair, purple, opposing
And the broad fields beneath them turn crimson,
Then howl I my heart nigh mad with rejoicing.

II

In hot summer have I great rejoicing
When the tempests kill the earth’s foul peace,
And the light’nings from black heav’n flash crimson,
And the fierce thunders roar me their music
And the winds shriek through the clouds mad, opposing,
And through all the riven skies God’s swords clash.

III

Hell grant soon we hear again the swords clash!
And the shrill neighs of destriers in battle rejoicing,
Spiked breast to spiked breast opposing!
Better one hour’s stour than a year’s peace
With fat boards, bawds, wine and frail music!
Bah! there’s no wine like the blood’s crimson!

IV

And I love to see the sun rise blood-crimson.
And I watch his spears through the dark clash
And it fills all my heart with rejoicing
And prys wide my mouth with fast music
When I see him so scorn and defy peace,
His lone might ’gainst all darkness opposing.

V

The man who fears war and squats opposing
My words for stour, hath no blood of crimson
But is fit only to rot in womanish peace
Far from where worth’s won and the swords clash
For the death of such sluts I go rejoicing;
Yea, I fill all the air with my music.

VI

Papiols, Papiols, to the music!
There’s no sound like to swords swords opposing,
No cry like the battle’s rejoicing
When our elbows and swords drip the crimson
And our charges ’gainst “The Leopard’s” rush clash.
May God damn for ever all who cry “Peace!”

VII

And let the music of the swords make them crimson
Hell grant soon we hear again the swords clash!
Hell blot black for always the thought “Peace”!



References:

[1] http://en.wikipedia.org/wiki/Sestina

[2] Richard Dore and Anton Geraschenko,”Sestinas and Primes” posted to http://stacky.net/wiki/index.php?title=Course_notes, and http://math.berkeley.edu/~anton/written/sestina.pdf

[3] Ezra Pound, “Sestina: Altaforte” (1909), (originally published int the English Review, 1909)

[4] John Bullitt, N. J. A. Sloane and J. H. Conway , http://oeis.org/A019567