The truncated tetrahedron covers the tetrahedron

At first, you might think this is obvious – just “clip” off each corner of the tetrahedron \Gamma_1 to create the truncated tetrahedron \Gamma_2 (by essentially creating a triangle from each of these clipped corners – see below for the associated graph). Then just map each such triangle to the corresponding vertex of the tetrahedron. No, it’s not obvious because the map just described is not a covering. This post describes one way to think about how to construct any covering.

First, color the vertices of the tetrahedron in some way.

\Gamma_1

The coloring below corresponds to a harmonic morphism \phi : \Gamma_2\to \Gamma_1:

\Gamma_2

All others are obtained from this by permuting the colors. They are all covers of \Gamma_1 = K_4 – with no vertical multiplicities and all horizontal multiplicities equal to 1. These 24 harmonic morphisms of \Gamma_2\to\Gamma_1 are all coverings and there are no other harmonic morphisms.

A footnote to Robert H. Mountjoy

In an earlier post titled Mathematical romantic? I mentioned some papers I inherited of one of my mathematical hero’s Andre Weil with his signature. In fact, I was fortunate enough to go to dinner with him once in Princeton in the mid-to-late 1980s – a very gentle, charming person with a deep love of mathematics. I remember he said he missed his wife, Eveline, who passed away in 1986. (They were married in 1937.)

All this is simply to motivate the question, why did I get these papers? First, as mentioned in the post, I was given Larry Goldstein‘s old office and he either was kind enough to gift me his old preprints or left them to be thrown away by the next inhabitant of his office. BTW, if you haven’t heard of him, Larry was a student of Shimura, when became a Gibbs Fellow at Yale, then went to the University of Maryland at COllege Park in 1969. He wrote lots of papers (and books) on number theory, eventually becoming a full professor, but eventually settled into computers and data science work. He left the University of Maryland about the time I arrived in the early 1980s to create some computer companies that he ran.

This motivates the question: How did Larry get these papers of Weil? I think Larry inherited them from Mountjoy (who died before Larry arrived at UMCP, but more on him later). This motivates the question, who is Mountjoy and how did he get them?

I’ve done some digging around the internet and here’s what I discovered.

The earliest mention I could find is when he was listed as a recipient of an NSF Fellowship in “Annual Report of the National Science Foundation: 1950-1953” under Chicago, Illinois, Mathematics, 1953. So he was a grad student at the University of Chicago in 1953. Andre Weil was there at the time. (He left sometime in 1958.) Mountjoy could have gotten the notes of Andre Weil then. Just before Weil left Chicago, Walter Lewis Baily arrived (in 1957, to be exact). This is important because in May 1965 the Notices of the AMS reported that reported:

Mountjoy, Robert Harbison
Abelian varieties attached to representations of discontinuous groups (S. Mac Lane and W. L. Baily)

(His thesis was published posthumously in American Journal of Mathematics Vol. 89 (1967)149-224.) This thesis is in a field studied by Weil and Baily but not Saunders.

But we’re getting ahead of ourselves. The 1962 issue of Maryland Magazine had this:


Mathematics Grant
A team of University of Maryland mathematics researchers have received a grant of $53,000 from the National Science Foundation to continue some technical investigations they started two years ago.
The mathematical study they are directing is entitled “Problems in Geometric Function Theory.” The project is under the direction of Dr. James Hummel. Dr. Mischael Zedek. and Prof. Robert H. Mountjoy, all of the Mathematics Department. They are assisted by four graduate-student researchists. The $53,000 grant is a renewal of an original grant which was made two years ago.

We know he was working at UMCP in 1962. 

Here’s the sad news. 

The newspaper Democrat and Chronicle, from Rochester, New York, on Wednesday, May 25, 1965 (Page 40) published the news that Robert H. Mountjoy “Died suddenly at Purcellville, VA, May 23, 1965”. I couldn’t read the rest (it’s behind a paywall but I could see that much). The next day, they published more: “Robert H. Mountjoy, son-in-law of Mr and Mrs Allen P Mills of Brighton, was killed in a traffic crash in Virgina. Mountjoy, about 30, a mathematics instructors at the University of Maryland, leaves a widow Sarah Mills Mountjoy and a 5-month old son Alexander, and his parents Mr and Mrs Lucius Mountjoy of Chicago.”

It’s so sad. The saying goes “May his memory be a blessing.” I never met him, but from what I’ve learned of Mountjoy, his memory is indeed a blessing.

The Riemann-Hurwitz formula for regular graphs

A little over 10 years ago, M. Baker and S. Norine (I’ve also seen this name spelled Norin) wrote a terrific paper on harmonic morphisms between simple, connected graphs (see “Harmonic morphisms and hyperelliptic graphs” – you can find a downloadable pdf on the internet of you google for it). Roughly speaking, a harmonic function on a graph is a function in the kernel of the graph Laplacian. A harmonic morphism between graphs is, roughly speaking, a map from one graph to another that preserves harmonic functions.

They proved quite a few interesting results but one of the most interesting, I think, is their graph-theoretic analog of the Riemann-Hurwitz formula. We define the genus of a simple connected graph \Gamma = (V,E) to be

{\rm genus}(\Gamma) = |E| - |V | + 1.


It represents the minimum number of edges that must be removed from the graph to make it into a tree (so, a tree has genus 0).

Riemann-Hurwitz formula (Baker and Norine): Let \phi:\Gamma_2\to \Gamma_1 be a harmonic morphism from a graph \Gamma_2 = (V_2,E_2) to a graph \Gamma_1 = (V_1, E_1). Then

{\rm genus}(\Gamma_2)-1 = {\rm deg}(\phi)({\rm genus}(\Gamma_1)-1)+\sum_{x\in V_2} [m_\phi(x)+\frac{1}{2}\nu_\phi(x)-1].

I’m not going to define them here but m_\phi(x) denotes the horizontal multiplicity and \nu_\phi(x) denotes the vertical multiplicity.

I simply want to record a very easy corollary to this, assuming \Gamma_2 = (V_2,E_2) is k_2-regular and \Gamma_1 = (V_1, E_1) is k_1-regular.

Corollary: Let \Gamma_2 \rightarrow \Gamma_1 be a non-trivial harmonic morphism from a connected k_2-regular graph
to a connected k_1-regular graph.
Then

\sum_{x\in V_2}\nu_\phi(x) = k_2|V_2| - k_1|V_1|\deg (\phi).

A table of small quartic graphs

This page is modeled after the handy wikipedia page Table of simple cubic graphs of “small” connected 3-regular graphs, where by small I mean at most 11 vertices.

These graphs are obtained using the SageMath command graphs(n, [4]*n), where n = 5,6,7,… .

5 vertices: Let V=\{0,1,2,3,4\} denote the vertex set. There is (up to isomorphism) exactly one 4-regular connected graphs on 5 vertices. By Ore’s Theorem, this graph is Hamiltonian. By Euler’s Theorem, it is Eulerian.
4reg5a: The only such 4-regular graph is the complete graph \Gamma = K_5.
graph4reg5
We have

  • diameter = 1
  • girth = 3
  • If G denotes the automorphism group then G has cardinality 120 and is generated by (3,4), (2,3), (1,2), (0,1). (In this case, clearly, G = S_5.)
  • edge set: \{(0, 1), (0, 2), (0, 3), (0, 4), (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)\}

6 vertices: Let V=\{0,1,\dots, 5\} denote the vertex set. There is (up to isomorphism) exactly one 4-regular connected graphs on 6 vertices. By Ore’s Theorem, this graph is Hamiltonian. By Euler’s Theorem, it is Eulerian.
4reg6a: The first (and only) such 4-regular graph is the graph \Gamma having edge set: \{(0, 1), (0, 2), (0, 4), (0, 5), (1, 2), (1, 3), (1, 4), (2, 3), (2, 5), (3, 4), (3, 5), (4, 5)\}.
graph4reg6
We have

  • diameter = 2
  • girth = 3
  • If G denotes the automorphism group then G has cardinality 48 and is generated by (2,4), (1,2)(4,5), (0,1)(3,5).

7 vertices: Let V=\{0,1,\dots, 6\} denote the vertex set. There are (up to isomorphism) exactly 2 4-regular connected graphs on 7 vertices. By Ore’s Theorem, these graphs are Hamiltonian. By Euler’s Theorem, they are Eulerian.
4reg7a: The 1st such 4-regular graph is the graph \Gamma having edge set: \{(0, 1), (0, 3), (0, 5), (0, 6), (1, 2), (1, 3), (1, 5), (2, 3), (2, 4), (2, 6), (3, 4), (4, 5), (4, 6), (5, 6)\}. This is an Eulerian, Hamiltonian (by Ore’s Theorem), vertex transitive (but not edge transitive) graph.
graph4reg7a
We have

  • diameter = 2
  • girth = 3
  • If G denotes the automorphism group then G has cardinality 14 and is generated by (1,5)(2,4)(3,6), (0,1,3,2,4,6,5).

4reg7b: The 2nd such 4-regular graph is the graph \Gamma having edge set: \{(0, 1), (0, 3), (0, 4), (0, 6), (1, 2), (1, 5), (1, 6), (2, 3), (2, 4), (2, 6), (3, 4), (3, 5), (4, 5), (5, 6)\}. This is an Eulerian, Hamiltonian graph (by Ore’s Theorem) which is neither vertex transitive nor edge transitive.
graph4reg7b
We have

  • diameter = 2
  • girth = 3
  • If G denotes the automorphism group then G has cardinality 48 and is generated by (3,4), (2,5), (1,3)(4,6), (0,2)

8 vertices: Let V=\{0,1,\dots, 7\} denote the vertex set. There are (up to isomorphism) exactly six 4-regular connected graphs on 8 vertices. By Ore’s Theorem, these graphs are Hamiltonian. By Euler’s Theorem, they are Eulerian.
4reg8a: The 1st such 4-regular graph is the graph \Gamma having edge set: \{(0, 1), (0, 5), (0, 6), (0, 7), (1, 3), (1, 6), (1, 7), (2, 3), (2, 4), (2, 5), (2, 7), (3, 4), (3, 6), (4, 5), (4, 6), (5, 7)\}. This is vertex transitive but not edge transitive.
graph4reg8a
We have

  • diameter = 2
  • girth = 3
  • If G denotes the automorphism group then G has cardinality 16 and is generated by (1,7)(2,3)(5,6) and (0,1)(2,4)(3,5)(6,7).

4reg8b: The 2nd such 4-regular graph is the graph \Gamma having edge set: \{(0, 1), (0, 5), (0, 6), (0, 7), (1, 3), (1, 6), (1, 7), (2, 3), (2, 4), (2, 5), (2, 7), (3, 4), (3, 6), (4, 5), (4, 6), (5, 7)\}. This is a vertex transitive (but not edge transitive) graph.
graph4reg8b
We have

  • diameter = 2
  • girth = 3
  • If G denotes the automorphism group then G has cardinality 48 and is generated by (2,3)(5,7), (1,3)(4,5), (0,1,3)(4,5,6), (0,4)(1,6)(2,5)(3,7).

4reg8c: The 3rd such 4-regular graph is the graph \Gamma having edge set: \{(0, 1), (0, 2), (0, 5), (0, 6), (1, 3), (1, 4), (1, 7), (2, 3), (2, 4), (2, 7), (3, 5), (3, 6), (4, 5), (4, 6), (5, 7), (6, 7)\}. This is a strongly regular (with “trivial” parameters (8, 4, 0, 4)), vertex transitive, edge transitive graph.
graph4reg8c
We have

  • diameter = 2
  • girth = 4
  • If G denotes the automorphism group then G has cardinality 1152=2^7\cdot 3^2 and is generated by (5,6), (4,7), (3,4), (2,5), (1,2), (0,1)(2,3)(4,5)(6,7).

4reg8d: The 4th such 4-regular graph is the graph \Gamma having edge set: \{(0, 1), (0, 2), (0, 4), (0, 6), (1, 3), (1, 5), (1, 6), (2, 3), (2, 5), (2, 7), (3, 4), (3, 7), (4, 5), (4, 6), (5, 7), (6, 7)\}. This graph is not vertex transitive, nor edge transitive.
graph4reg8d
We have

  • diameter = 2
  • girth = 3
  • If G denotes the automorphism group then G has cardinality 16 and is generated by (3,5), (1,4), (0,2)(1,3)(4,5)(6,7), (0,6)(2,7).

4reg8e: The 5th such 4-regular graph is the graph \Gamma having edge set: \{(0, 1), (0, 2), (0, 6), (0, 7), (1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 7), (3, 5), (3, 7), (4, 5), (4, 6), (5, 6), (6, 7)\}. This graph is not vertex transitive, nor edge transitive.
graph4reg8e
We have

  • diameter = 2
  • girth = 3
  • If G denotes the automorphism group then G has cardinality 4 and is generated by (0,1)(2,4)(3,6)(5,7), (0,2)(1,4)(3,6).

4reg8f: The 6th (and last) such 4-regular graph is the bipartite graph \Gamma=K_{4,4} having edge set: \{(0, 1), (0, 2), (0, 6), (0, 7), (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 5), (3, 7), (4, 5), (4, 6), (5, 6), (5, 7), (6, 7)\}. This graph is not vertex transitive, nor edge transitive.
graph4reg8f
We have

  • diameter = 2
  • girth = 3
  • If G denotes the automorphism group then G has cardinality 12 and is generated by (3,4)(6,7), (1,2), (0,3)(5,6).

9 vertices: Let V=\{0,1,\dots, 8\} denote the vertex set. There are (up to isomorphism) exactly 16 4-regular connected graphs on 9 vertices. Perhaps the most interesting of these is the strongly regular graph with parameters (9, 4, 1, 2) (also distance regular, as well as vertex- and edge-transitive). It has an automorphism group of cardinality 72, and is referred to as d4reg9-14 below.

Without going into details, it is possible to theoretically prove that there are no harmonic morphisms from any of these graphs to either the cycle graph C_4 or the complete graph K_4. However, both d4reg9-3 and d4reg9-14 not only have harmonic morphisms to C_3, they each may be regarded as a multicover of C_3.

d4reg9-1
Gamma edges: E1 = [(0, 1), (0, 2), (0, 7), (0, 8), (1, 2), (1, 3), (1, 7), (2, 3), (2, 8), (3, 4), (3, 5), (4, 5), (4, 6), (4, 8), (5, 6), (5, 7), (6, 7), (6, 8)] 
diameter:  2 
girth:  3 
is connected:  True 
aut gp size:  12 
aut gp gens:  [(1,2)(4,5)(7,8), (0,1)(3,8)(5,6), (0,4)(1,5)(2,6)(3,7)] 

d4reg9_1

d4reg9-2 
Gamma edges: E1 = [(0, 1), (0, 3), (0, 7), (0, 8), (1, 2), (1, 3), (1, 7), (2, 3), (2, 5), (2, 8), (3, 4), (4, 5), (4, 6), (4, 8), (5, 6), (5, 7), (6, 7), (6, 8)] 
diameter:  2 
girth:  3 
is connected:  True 
aut gp size:  2 
aut gp gens:  [(0,5)(1,6)(2,8)(3,4)] 

d4reg9_2

d4reg9-3 
Gamma edges: E1 = [(0, 1), (0, 2), (0, 7), (0, 8), (1, 2), (1, 3), (1, 4), (2, 3), (2, 8), (3, 4), (3, 5), (4, 5), (4, 6), (5, 6), (5, 7), (6, 7), (6, 8), (7, 8)] 
diameter:  2 
girth:  3 
is connected:  True 
aut gp size:  18 
aut gp gens:  [(1,7)(2,8)(3,6)(4,5), (0,1,4,6,8,2,3,5,7)] 

d4reg9_3

d4reg9-4 
Gamma edges: E1 = [(0, 1), (0, 5), (0, 7), (0, 8), (1, 2), (1, 4), (1, 7), (2, 3), (2, 4), (2, 5), (3, 4), (3, 6), (3, 8), (4, 5), (5, 6), (6, 7), (6, 8), (7, 8)] 
diameter:  2 
girth:  3 
is connected:  True 
aut gp size:  4 
aut gp gens:  [(2,4), (0,6)(1,3)(7,8)] 

d4reg9_4

d4reg9-5 
Gamma edges: E1 = [(0, 1), (0, 3), (0, 5), (0, 8), (1, 2), (1, 4), (1, 7), (2, 3), (2, 5), (2, 7), (3, 4), (3, 8), (4, 5), (4, 6), (5, 6), (6, 7), (6, 8), (7, 8)] 
diameter:  2 
girth:  3 
is connected:  True 
aut gp size:  12 
aut gp gens:  [(1,5)(2,4)(6,7), (0,1)(2,3)(4,5)(7,8)] 

d4reg9_5

d4reg9-6 
Gamma edges: E1 = [(0, 1), (0, 3), (0, 7), (0, 8), (1, 2), (1, 5), (1, 6), (2, 3), (2, 5), (2, 6), (3, 4), (3, 8), (4, 5), (4, 7), (4, 8), (5, 6), (6, 7), (7, 8)] 
diameter:  2 
girth:  3 
is connected:  True 
aut gp size:  8 
aut gp gens:  [(2,6)(3,7), (0,3)(1,2)(4,7)(5,6)] 

d4reg9_6

d4reg9-7 
Gamma edges: E1 = [(0, 1), (0, 3), (0, 4), (0, 8), (1, 2), (1, 3), (1, 6), (2, 3), (2, 5), (2, 7), (3, 4), (4, 5), (4, 8), (5, 6), (5, 7), (6, 7), (6, 8), (7, 8)] 
diameter:  2 
girth:  3 
is connected:  True 
aut gp size:  2 
aut gp gens:  [(0,3)(1,4)(2,8)(5,6)] 

d4reg9_7

d4reg9-8 
Gamma edges: E1 = [(0, 1), (0, 3), (0, 7), (0, 8), (1, 2), (1, 3), (1, 6), (2, 3), (2, 5), (2, 7), (3, 4), (4, 5), (4, 6), (4, 8), (5, 6), (5, 8), (6, 7), (7, 8)] 
diameter:  2 
girth:  3 
is connected:  True 
aut gp size:  2 
aut gp gens:  [(0,8)(1,5)(2,6)(3,4)] 

d4reg9_8

d4reg9-9 
Gamma edges: E1 = [(0, 1), (0, 3), (0, 6), (0, 8), (1, 2), (1, 3), (1, 6), (2, 3), (2, 5), (2, 7), (3, 4), (4, 5), (4, 7), (4, 8), (5, 6), (5, 8), (6, 7), (7, 8)] 
diameter:  2 
girth:  3 
is connected:  True 
aut gp size:  4 
aut gp gens:  [(5,7), (0,3)(2,6)(4,8)] 

d4reg9_9

d4reg9-10 
Gamma edges: E1 = [(0, 1), (0, 3), (0, 5), (0, 8), (1, 2), (1, 4), (1, 6), (2, 3), (2, 5), (2, 7), (3, 4), (3, 7), (4, 5), (4, 8), (5, 6), (6, 7), (6, 8), (7, 8)] 
diameter:  2 
girth:  3 
is connected:  True 
aut gp size:  16 
aut gp gens:  [(2,6)(3,8), (1,5), (0,1)(2,3)(4,5)(6,8)] 

d4reg9_10

d4reg9-11 
Gamma edges: E1 = [(0, 1), (0, 3), (0, 7), (0, 8), (1, 2), (1, 4), (1, 6), (2, 3), (2, 5), (2, 7), (3, 4), (3, 5), (4, 5), (4, 8), (5, 6), (6, 7), (6, 8), (7, 8)] 
diameter:  2 
girth:  3 
is connected:  True 
aut gp size:  8 
aut gp gens:  [(2,4)(7,8), (0,2)(3,7)(4,6)(5,8)] 

d4reg9_11

d4reg9-12 
Gamma edges: E1 = [(0, 1), (0, 3), (0, 6), (0, 8), (1, 2), (1, 4), (1, 6), (2, 3), (2, 5), (2, 7), (3, 4), (3, 7), (4, 5), (4, 8), (5, 6), (5, 8), (6, 7), (7, 8)] 
diameter:  2 
girth:  3 
is connected:  True 
aut gp size:  18 
aut gp gens:  [(1,6)(2,5)(3,8)(4,7), (0,1,6)(2,7,3)(4,5,8), (0,2)(1,3)(5,8(6,7)] 

d4reg9_12

d4reg9-13 
Gamma edges: E1 = [(0, 1), (0, 3), (0, 4), (0, 8), (1, 2), (1, 5), (1, 6), (2, 3), (2, 5), (2, 7), (3, 4), (3, 7), (4, 5), (4, 8), (5, 6), (6, 7), (6, 8), (7, 8)] 
diameter:  2 
girth:  3 
is connected:  True 
aut gp size:  8 
aut gp gens:  [(2,6)(3,8), (0,1)(2,3)(4,5)(6,8), (0,4)(1,5)] 

d4reg9_13

d4reg9-14 
Gamma edges: E1 = [(0, 1), (0, 3), (0, 4), (0, 8), (1, 2), (1, 5), (1, 8), (2, 3), (2, 5), (2, 7), (3, 4), (3, 7), (4, 5), (4, 6), (5, 6), (6, 7), (6, 8), (7, 8)] 
diameter:  2 
girth:  3 
is connected:  True 
aut gp size:  72 
aut gp gens:  [(2,5)(3,4)(6,7), (1,3)(4,8)(5,7), (0,1)(2,3)(4,5)] 

d4reg9_14

d4reg9-15 
Gamma edges: E1 = [(0, 1), (0, 4), (0, 6), (0, 8), (1, 2), (1, 3), (1, 5), (2, 3), (2, 4), (2, 7), (3, 4), (3, 7), (4, 5), (5, 6), (5, 8), (6, 7), (6, 8), (7, 8)] 
diameter:  2 
girth:  3 
is connected:  True 
aut gp size:  32 
aut gp gens:  [(6,8), (2,3), (1,4), (0,1)(2,6)(3,8)(4,5)] 

d4reg9_15

d4reg9-16 
Gamma edges: E1 = [(0, 1), (0, 3), (0, 7), (0, 8), (1, 2), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5), (3, 7), (3, 8), (4, 5), (4, 6), (5, 6), (6, 7), (6, 8), (7, 8)] 
diameter:  2 
girth:  3 
is connected:  True 
aut gp size:  16 
aut gp gens:  [(7,8), (4,5), (0,1)(2,3)(4,7)(5,8), (0,2)(1,3)(4,7)(5,8)] 

d4reg9_16

10 vertices: Let V=\{0,1,\dots, 9\} denote the vertex set. There are (up to isomorphism) exactly 59 4-regular connected graphs on 10 vertices. One of these actually has an automorphism group of cardinality 1. According to SageMath: Only three of these are vertex transitive, two (of those 3) are symmetric (i.e., arc transitive), and only one (of those 2) is distance regular.

Example 1: The quartic, symmetric graph on 10 vertices that is not distance regular is depicted below. It has diameter 2, girth 4, chromatic number 3, and has an automorphism group of order 320 generated by \{(7,8), (4,6), (1,2), (1,7)(2,8)(3,4)(5,6), (0,1,3,4,7)(2,5,6,8,9)\}.

d4reg10-46a

Example 2: The quartic, distance regular, symmetric graph on 10 vertices is depicted below. It has diameter 3, girth 4, chromatic number 2, and has an automorphism group of order 240 generated by \{(2,5)(4,7), (2,8)(3,4), (1,5)(7,9), (0,1,3,2,7,6,9,8,4,5)\}.

d4reg10-51a

11 vertices: There are (up to isomorphism) exactly 265 4-regular connected graphs on 11 vertices. Only two of these are vertex transitive. None are distance regular or edge transitive.

Example 1: One of the vertex transitive graphs is depicted below. It has diameter 2, girth 4, chromatic number 3, and has an automorphism group of order 22 generated by \{(1,10)(2,9)(3,4)(5,6)(7,8), (0,1,5,4,2,7,8,9,3,6,10)\}.

Example 2:The second vertex transitive graph is depicted below. It has diameter 3, girth 3, chromatic number 4, and has an automorphism group of order 22 generated by \{(1,5)(2,7)(3,6)(4,8)(9,10), (0,1,3,2,4,10,9,8,7,6,5)\}.

NCF Boolean functions

I recently learned about a new class of seemingly complicated, but in fact very simple functions which are called by several names, but perhaps most commonly as NCF Boolean functions (NCF is an abbreviation for “nested canalyzing function,” a term used by mathematical biologists). These functions were independently introduced by theoretical computer scientists in the 1960s using the term unate cascade functions. As described in [JRL2007] and [LAMAL2013], these functions have applications in a variety of scientific fields. This post describes these functions.

A Boolean function of n variables is simply a function f:GF(2)^n\to GF(2). Denote the GF(2)-vector space of such functions by B(n). We write an element of this space as f(x_1,x_2,\dots,x_n), where the variables x_i will be called coordinate variables. Let
Res_{x_i=a}:B(n)\to B(n-1)
denote the restriction map sending f(x_1,x_2,\dots,x_n) to f(x_1,x_2,\dots,x_{i-1},a,x_{i+1},\dots, x_n). In this post, the cosets
H_{i,a,n}=\{x=(x_1,x_2,\dots,x_n) \in GF(2)^n\ |\ x_i=a\}
will be called coordinate hyperplanes (a \in GF(2), 1\leq i\leq n). A function in B(n) which is constant along some coordinate hyperplane is called canalyzing. An NCF function is a function f\in B(n) which (a) is constant along some coordinate hyperplane H_{i_1,a_1,n}, (b) whose restriction f_1 = Res_{x_{i_1}=a_1}(f)\in B(n-1) is constant along some coordinate hyperplane H_{i_2,a_2,n-1}\subset GF(2)^{n-1}, (c) whose restriction f_2 = Res_{x_{i_2}=a_2}(f_1)\in B(n-2) is constant along some coordinate hyperplane H_{i_2,a_2,n-2}\subset GF(2)^{n-2}, (d) and so on. This “nested” inductive definition might seem complicated, but to a computer it’s pretty simple and, to boot, it requires little memory to store.

If 1\leq i\leq n and x=(x_1,x_2,\dots,x_n) \in GF(2)^n then let x^i\in GF(2)^n denote the vector whose i-th coordinate is flipped (bitwise). The sensitivity of f\in B(n) at x is
s(f,x) = |\{i\ |\ 1\leq i\leq n, f(x)\not= f(x^i)\}|. Roughly speaking, it’s the number of single-bit changes in x that change the value of f(x). The (maximum) sensitivity is the quantity
s(f)=max_x s(f,x). The block sensitivity is defined similarly, but you allow blocks of indices of coordinates to by flipped bitwise, as opposed to only one. It’s possible to

  • compute the sensitivity of any NCF function,
  • show the block sensitivity is equal to the sensitivity,
  • compute the cardinality of the set of all monotone NCF functions.

For details, see for example Li and Adeyeye [LA2012].

REFERENCES
[JRL2007] A.S. Jarrah, B. Raposa, R. Laubenbachera, “Nested Canalyzing, Unate Cascade, and Polynomial Functions,” Physica D. 2007 Sep 15; 233(2): 167–174.

[LA2012] Y. Li, J.O. Adeyeye, “Sensitivity and block sensitivity of nested canalyzing function,” ArXiV 2012 preprint. (A version of this paper was published later in Theoretical Comp. Sci.)

[LAMAL2013] Y. Li, J.O. Adeyeye, D. Murrugarra, B. Aguilar, R. Laubenbacher, “Boolean nested canalizing functions: a comprehensive analysis,” ArXiV, 2013 preprint.

Differential equations and SageMath

The files below were on my teaching page when I was a college teacher. Since I retired, they disappeared. Samuel Lelièvre found an archived copy on the web, so I’m posting them here.

The files are licensed under the Attribution-ShareAlike Creative Commons license.

  1. Partial fractions handout, pdf
  2. Introduction to matrix determinants handout, pdf
  3. Impulse-response handout, pdf
  4. Introduction to ODEs, pdf
  5. Initial value problems, pdf
  6. Existence and uniqueness, pdf
  7. Euler’s method for numerically approximating solutions to DEs, pdf.
    Includes both 1st order DE case (with Euler and improved Euler) and higher order DE and systems of DEs cases, without improved Euler.
  8. Direction fields and isoclines, pdf
  9. 1st order ODEs, separable and linear cases, pdf
  10. A falling body problem in Newtonian mechanics, pdf
  11. A mixing problem, pdf
  12. Linear ODEs, I, pdf
  13. Linear ODEs, II, pdf
  14. Undetermined coefficients for non-homogeneous 2nd order constant coefficient ODEs, pdf
  15. Variation of parameters for non-homogeneous 2nd order constant coefficient ODEs, pdf.
  16. Annihilator method for non-homogeneous 2nd order constant coefficient ODEs, pdf.
    I found students preferred (the more-or-less equivalent) undetermined coefficient method, so didn’t put much effort into these notes.
  17. Springs, I, pdf
  18. Springs, II, pdf
  19. Springs, III, pdf
  20. LRC circuits, pdf
  21. Power series methods, I, pdf
  22. Power series methods, II, pdf
  23. Introduction to Laplace transform methods, I, pdf
  24. Introduction to Laplace transform methods, II, pdf
  25. Lanchester’s equations modeling the battle between two armies, pdf
  26. Row reduction/Gauss elimination method for systems of linear equations, pdf.
  27. Eigenvalue method for homogeneous constant coefficient 2×2 systems of 1st order ODEs, pdf.
  28. Variation of parameters for first order non-homogeneous linear constant coefficient systems of ODEs, pdf.
  29. Electrical networks using Laplace transforms, pdf
  30. Separation of variables and the Transport PDE, pdf
  31. Fourier series, pdf.
  32. one-dimensional heat equation using Fourier series, pdf.
  33. one-dimensional wave equation using Fourier series, pdf.
  34. one-dimensional Schroedinger’s wave equation for a “free particle in a box” using Fourier series, pdf.
  35. All these lectures collected as one pdf (216 pages).
    While licensed Attribution-ShareAlike CC, in the US this book is in the public domain, as it was written while I was a US federal government employee as part of my official duties. A warning – it has lots of typos. The latest version, written with Marshall Hampton, is a JHUP book, much more polished, available on amazon and the JHUP website. Google “Introduction to Differential Equations Using Sage”.

Course review: pdf

Love, War, and Zombies, pdf
This set of slides is of a lecture I would give if there was enough time towards the end of the semester

Integral Calculus and SageMath

Long ago, using LaTeX I assembled a book on Calculus II (integral calculus), based on notes of mine, Dale Hoffman (which was written in word), and William Stein. I ran out of energy to finish it and the source files mostly disappeared from my HD. Recently, Samuel Lelièvre found a copy of the pdf of this book on the internet (you can download it here). It’s licensed under a Creative Commons Attribution 3.0 United States License. You are free to print, use, mix or modify these materials as long as you credit the original authors.

Table of Contents

0 Preface

1 The Integral
1.1 Area
1.2 Some applications of area
1.2.1 Total accumulation as “area”
1.2.2 Problems
1.3 Sigma notation and Riemann sums
1.3.1 Sums of areas of rectangles
1.3.2 Area under a curve: Riemann sums
1.3.3 Two special Riemann sums: lower and upper sums
1.3.4 Problems
1.3.5 The trapezoidal rule
1.3.6 Simpson’s rule and Sage
1.3.7 Trapezoidal vs. Simpson: Which Method Is Best?
1.4 The definite integral
1.4.1 The Fundamental Theorem of Calculus
1.4.2 Problems
1.4.3 Properties of the definite integral
1.4.4 Problems
1.5 Areas, integrals, and antiderivatives
1.5.1 Integrals, Antiderivatives, and Applications
1.5.2 Indefinite Integrals and net change
1.5.3 Physical Intuition
1.5.4 Problems
1.6 Substitution and Symmetry
1.6.1 The Substitution Rule
1.6.2 Substitution and definite integrals
1.6.3 Symmetry
1.6.4 Problems

2 Applications
2.1 Applications of the integral to area
2.1.1 Using integration to determine areas
2.2 Computing Volumes of Surfaces of Revolution
2.2.1 Disc method
2.2.2 Shell method
2.2.3 Problems
2.3 Average Values
2.3.1 Problems
2.4 Moments and centers of mass
2.4.1 Point Masses
2.4.2 Center of mass of a region in the plane
2.4.3 x-bar For A Region
2.4.4 y-bar For a Region
2.4.5 Theorems of Pappus
2.5 Arc lengths
2.5.1 2-D Arc length
2.5.2 3-D Arc length

3 Polar coordinates and trigonometric integrals
3.1 Polar Coordinates
3.2 Areas in Polar Coordinates
3.3 Complex Numbers
3.3.1 Polar Form
3.4 Complex Exponentials and Trigonometric Identities
3.4.1 Trigonometry and Complex Exponentials
3.5 Integrals of Trigonometric Functions
3.5.1 Some Remarks on Using Complex-Valued Functions

4 Integration techniques
4.1 Trigonometric Substitutions
4.2 Integration by Parts
4.2.1 More General Uses of Integration By Parts
4.3 Factoring Polynomials
4.4 Partial Fractions
4.5 Integration of Rational Functions Using Partial Fractions
4.6 Improper Integrals
4.6.1 Convergence, Divergence, and Comparison

5 Sequences and Series
5.1 Sequences
5.2 Series
5.3 The Integral and Comparison Tests
5.3.1 Estimating the Sum of a Series
5.4 Tests for Convergence
5.4.1 The Comparison Test
5.4.2 Absolute and Conditional Convergence
5.4.3 The Ratio Test
5.4.4 The Root Test
5.5 Power Series
5.5.1 Shift the Origin
5.5.2 Convergence of Power Series
5.6 Taylor Series
5.7 Applications of Taylor Series
5.7.1 Estimation of Taylor Series

6 Some Differential Equations
6.1 Separable Equations
6.2 Logistic Equation

7 Appendix: Integral tables

Problem of the Week, #121

A former colleague Bill Wardlaw (March 3, 1936-January 2, 2013) used to create a “Problem of the Week” for his students, giving a prize of a cookie if they could solve it. Here is one of them.

Problem 121

The Maryland “Big Game” lottery is played by selecting 5 different numbers in \{ 1,2,3,\dots, 50\} and then selecting one of the numbers in \{ 1,2,3,\dots, 36\}. The first section is an unordered selection without replacement (so, arrange them in increasing order if you like) but the second selection can repeat one of the 5 numbers initially picked.

How many ways can this be done?

Questions about quadratic residues

Let P denote the set of all primes and, for p \in P, let (*/p) denote the Legendre quadratic residue symbol mod p. Let {\mathbb N}=\{1,2,\dots\} denote the set of natural numbers and let

L: {\mathbb N}\to \{-1,0,1\}^P,

denote the mapping L(n)=( (n/2), (n/3), (n/5), \dots), so the kth component of L(n) is L(n)_k=(n/p_k) where p_k denotes the kth prime in P. The following result is “well-known” (or, as the joke goes, it’s well-known to those who know it well:-).

Theorem: The restriction of L to the subset {\mathbb S} of square-free integers is an injection.

In fact, one can be a little more precise. Let P_{\leq M} denote the set of the first M primes, let {\mathbb S}_N denote the set of square-free integers \leq N, and let

L_M: {\mathbb N}\to \{-1,0,1\}^{P_M},

denote the mapping L_M(n)=( (n/2), (n/3), (n/5), \dots, (n/p_M)).

Theorem: For each N>1, there is an M=M(N)>1 such that the restriction of L_M to the subset {\mathbb S}_N is an injection.

I am no expert in this field, so perhaps the following question is known.

Question: Can one give an effective upper bound on M=M(N)>1 as a function of N>1?

I’ve done some computer computations using SageMath and it seems to me that

M=O(N)

(i.e., there is a linear bound) is possible. On the other hand, my computations were pretty limited, so that’s just a guess.

Dodecahedral Faces of M12

 

Dodecahedral Faces of M12

 

by Ann Luers Casey

 

This post constitutes part of the math honors thesis written in spring 1997 at the USNA, advised by David Joyner. It is in the public domain.

Groups are objects in mathematics that measure symmetry in nature. A group is a set with a binary operation that has an inverse, an identity and is associative. For example, a clock has 12-fold symmetry. A more unusual group is a sporadic, non-abelian simple group. It can be very interesting to look more closely at such a group that arises naturally. One such group is M12. This post explores two different ways of creating M12 and then looks at twelve different ways M12 appears in mathematics, hence the pun the “dodecahedral faces” in the title. Specifically, this post relates M12 to the Mongean shuffle, hexads of a Steiner system, Golay codes, the Hadamard matrix of order 12, 5-transitivity, presentations, crossing the Rubicon, the minimog, the kitten, mathematical blackjack, sporadic groups, and the stabilizer in M24 of a dodecad.

Definitions:

Homomorphism: Let G1, G2 be groups with *1 denoting the group operation for G1 and *2 the group operation for G2. A function f : G1–>G2 is a homomorphism if and only if for all a,b, in Gwe have

f(a *1 b) = f(a) *2 f(b).

Isomorphism: If a homomorphism is bijective, then it is called an
isomorphism.

Automorphism: An isomorphism from a group G to itself is an automorphism.

Notation:

  • Let Fq denote the finite field with q elements, q is a power of a prime.
  • Z = the invertible scalar 2×2 matrices with entries in Fqx.
  • Let PGL2(Fq) = GL2(Fq)/Z = {A*Z | A is in GL2(Fq)}, with multiplication given by
    (A*Z)(B*Z) = (A*B)Z. This is the projective linear group over Fq.
  • LF(Fq) is the group of linear fractional transformations x–>(ax+b)/(cx+d).

Claim: There is a group theoretic isomorphism between PGL2(Fq) and LF(Fq). (See [11], Theorem 9.47 for a proof.)

Claim: LF(Fq) acts 3-transitively on the set P1(Fq) (q>3). I.e., one can send any triple to any other triple in P1(Fq) by using a suitable linear fractional transformation. (See [11], Theorem 9.48 for a proof.)

Theorem

PSL2(Fq) = < x–>x+1, x–>kx, x–>-1/x>, where k is any element in Fqthat generates the multiplicative group of squares.

For a proof, see [12], ch 10, section 1.

One way to construct the Mathieu group M12 is the following, accredited to Conway.

M12 = < PSL2(F11), (2 10)(3 4)(5 9)(6 7) >.More explicitly, let

  • f1 be a cyclic permutation = x–> x+1 = (0,1,2,…,10)(inf).
  • f2 = x–>kx = (0)(1 3 9 5 4)(2 6 7 10 8)(inf) when k=3.
  • f3 = x–>-1/x = (0 inf)(1 10)(2 5)(3 7)(4 8)(6 9).
  • f4 = (2 10)(3 4)(5 9)(6 7).

Then M12 = < f1, f2, f3, f4 >. Therefore, M12 is a subgroup of the symmetric group on 12
symbols, namely inf, 0, 1, …, 10.

Another way to construct M12 is given later under 5-transitivity.

There are many occurrences of M12 in mathematics, but here I will list and explain twelve of them:

  1. Mongean Shuffle
  2. Steiner Hexad
  3. Golay Code
  4. Hadamard Matrices
  5. 5- Transitivity
  6. Presentations
  7. Crossing the Rubicon
  8. M12 and the Minimog
  9. Kitten
  10. Mathematical Blackjack or Mathieu’s 21
  11. Sporadic Groups
  12. Stabilizer in M24 of a dodecad

    1. Mongean Shuffle

     

    The Mongean shuffle concerns a deck of twelve cards, labeled 0 through 11. The permutation

    r(t) = 11-t

    reverses the cards around. The permutation

    s(t) = min(2t,23-2t)

    is called the Mongean Shuffle. The permutation group M12 is generated by r and s: M12 = < r,s >, as a subgroup of S12. (See [12], Chap. 11, Sec. 17 or [18])

    2. Steiner Hexad

     

    Jacob Steiner (1796-1863) was a Swiss mathematician specializing in projective goemetry. (It is said that he did not learn to read or write until the age of 14 and only started attending school at the age of 18.) The origins of “Steiner systems” are rooted in problems of plane geometry.

    Let T be a given set with n elements. Then the Steiner system S(k,m,n) is a collection S = {S1, … ,Sr} of subsets of T such that

    • |Si| = m,
    • For any subset R in T with |R| = k there is a unique i, 1<=i<=n such that R is contained in Si. |S(k,m,n)| = (n choose k)/(m choose k).

    If any set H has cardinality 6 (respectively 8, 12) then H is called a hexad, (respectively octad, dodecad.)

    Let’s look at the Steiner System S(5,6,12) and M12. We want to construct the Steiner system S(5,6,12) using the projective line P1(F11). To define the hexads in the Steiner system, denote

    • the projective line over F11 by P1(F11)={inf,0,1,…,10}.
    • Q = {0,1,3,4,5,9}=the quadratic residues union 0
    • G = PSL2(F11)
    • S = set of all images of Q under G. (Each element g in G will send Q to a subset of P1(F11). )

    There are always six elements in such a hexad. There are 132 such hexads. If I know five of the elements in a hexad of S, then the sixth element is uniquely determined. Therefore S is a Steiner system of type (5,6,12).

    Theorem:
    M12 sends a hexad in a Steiner system to another hexad in a Steiner system. In fact, the automorphism group of a Steiner system of type (5,6,12) is isomorphic to M12.

    (For a proof, see [11], Theorem 9.78.)

    The hexads of S form a Steiner system of type (5,6,12), so

    M12 = < g in S12 | g(s) belongs to S, for all s in S > .

    In other words, M12 is the subgroup stabilizing S. The hexads support the weight six words of the Golay code, defined next. (For a proof, see  [6].)

    3. Golay Code

     

    ” The Golay code is probably the most important of all codes for both practical and theoretical reasons.” ([17], pg. 64)

    M. J. E. Golay (1902-1989) was a Swiss physicist known for his work in infrared spectroscopy among other things. He was one of the founding fathers of coding theory, discovering GC24 in 1949 and GC12 in 1954.

    A code C is a vector subspace of (Fq)for some n >=1 and some prime power q =pk.
    An automorphism of C is a vector space isomorphism, f:C–>C.

    If w is a code word in Fqn, n>1, then the number of non-zero coordinates of w is called the weight w, denoted by wt(w). A cyclic code is a code which has the property that whenever (c0,c1,…,cn-1) is a code word then so is (cn-1,c0,…,cn-2).
    If c=(c0,c1,…,cn-1) is a code word in a cyclic code C then we can associate to it a polynomial g_c(x)=c0 + c1x + … + cn-1xn-1. It turns out that there is a unique monic polynomial with coefficients in Fq

    of degree >1 which divides all such polynomials g_c(x). This polynomial is called
    a generator polynomial for C, denoted g(x).

    Let n be a positive integer relatively prime to q and let alpha be a primitive n-th root of unity. Each generator g of a cyclic code C of length n has a factorization of the form g(x) = (x-alphak1)… (x-alphakr), where {k1,…,kr} are in {0,…,n-1} [17]. The numbers alphaki, 1≤ i≤ r, are called the zeros of the code C.

    If p and n are distinct primes and p is a square mod n, then the quadratic residue code of length n over Fp is the cyclic code whose generator polynomial has zeros
    {alphak | k is a square mod n} [17]. The ternary Golary code GC11 is the quadratic
    residue code of length 11 over F3.

    The ternary Golay code GC12 is the quadratic residue code of length 12 over F3 obtained by appending onto GC11 a zero-sum check digit [12].

    Theorem:
    There is a normal subgroup N of Aut(GC12) of order 2 such that Aut(GC12)/N is isomorphic to M12. M12 is a quotient of Aut(GC12) by a subgroup or order 2. In other words, M12 fits into the following short exact sequence:

    1–>N–>Aut(GC12)–>M12–>1

    Where i is the embedding and N in Aut(GC12) is a subgroup of order 2. See [6].

    4. Hadamard Matrices

     

    Jacques Hadamard (1865-1963) was a French mathematician who did important work in analytic number theory. He also wrote a popular book “The psychology in invention in the mathematical field” (1945).

    A Hadamard matrix is any n x n matrix with a +1 or -1 in every entry such that the absolute value of the determinant is equal to nn/2.

    An example of a Hadamard matrix is the Paley-Hadamard matrix. Let p be a prime of the form 4N-1, p > 3. A Paley-Hadamard matrix has order p+1 and has only +1’s and -1’s as entries. The columns and rows are indexed as (inf,0,1,2,…,p-1). The infinity row and the infinity column are all +1’s. The zero row is -1 at the 0th column and at the columns that are quadratic non-residues mod p; the zero row is +1 elsewhere. The remaining p-1 rows are cyclic shifts of the finite part of the second row. For further details, see for example [14].

    When p = 11 this construction yields a 12×12 Hadamard matrix.

    Given two Hadamard Matrices A, B we call them left-equivalent if there is an nxn signed permutation matrix P such that PA = B.

    The set {P nxn signed permutation matrix| AP is left equivalent to A} is called the automorphism group of A. In other words, a matrix is an automorphism of the Hadamard matrix, if it is a nxn monomial matrix with entries in {0,+1,-1} and when it is multiplies the Hadamard matrix on the right, only the rows may be permuted, with a sign change in some rows allowed.

    Two nxn Hadamard matrices A, B are called equivalent if there are nxn signed permutation matrices P1, P2 such that A = P1 *B *P2.

    All 12×12 Hadamard matrices are equivalent ([13][16] pg. 24). The group of automorphisms of any 12×12 Hadamard matrix is isomorphic to the Mathieu group M12 ([14] pg 99).

     

    5. 5-Transitivity

     

    Emile Mathieu (1835-1890) was a mathematical physicist known for his solution to the vibrations of an elliptical membrane.

    The fact that M12 acts 5-transitively on a set with 12 elements is due to E. Mathieu who proved the result in 1861. (Some history may be found in [15].)

    There are only a finite number of types of 5-transitive groups, namely Sn (n>=5), An (n>=7), M12 and M24. (For a proof, see [11])

    Let G act on a set X via phi : G–>SX. G is k-transitive if for each pair of ordered k-tuples (x1, x2,…,xk), (y1,y2,…,yk), all xi and yi elements belonging to X, there exists a g in G such that yi = phi(g)(xi) for each i in {1,2,…,k}.

    M12 can also be constructed as in Rotman [11], using transitive extensions, as follows (this construction appears to be due originally to Witt). Let fa,b,c,d(x)=(ax+b)/(cx+d), let

    M10 = < fa,b,c,d, fa’,b’,c’,d’ |ad-bc is in Fqx, a’d’-b’c’ is not in Fqx >,

    q = 9.

    pi = generator of F9x, so that F9x = < pi> = C8.

    Using Thm. 9.51 from Rotman, we can create a transitive extension of M10. Let omega be a new symbol and define

    M11 = < M10, h| h = (inf, omega)(pi, pi2)(pi3,pi7) (pi5,pi6)>.

    Let P1(F9) = {inf, 0, 1, pi, pi2, … , pi7}. Then M11 is four transitive on Y0 = P1(F9) union {omega}, by Thm 9.51.

    Again using Thm. 9.51, we can create a transitive extension of M11. Let sigma be a new symbol and define

    M12 = < M11, k>, where k = (omega, sig)(pi,pi3) (pi2,pi6)(pi5,pi7). M12 is 5-transitive on Y1 = Y0 union {sig}, by Th. 9.51.

    Now that we constructed a particular group that is 5-transitive on a particular set with 12 elements, what happens if we have a group that is isomorphic to that group? Is this new group also 5-transitive?

    Let G be a subgroup of S12 isomorphic to the Mathieu group M12. Such a group was constructed in Section 1.

    Lemma: There is an action of G on the set {1,2,…,12} which is 5-transitive.

    proof: Let Sig : G –> M12 be an isomorphism. Define g(i) = Sig(g)(i), where i = {1,2,…,12}, g is in G. This is an action since Sig is an isomorphism. Sig-1(h)(i) = h(i) for all g in M12, i in Y1. Using some h in M12, any i1,…,i5

    in Y1 can be sent to any j1,…,j5 in Y1. That is, there exists an h in M12 such that h(ik) = jk, k= 1,…,5 since M12 is 5-transitive. Therefore, Sig-1(h)(ik) = jk = g(ik). This action is 5-transitive. QED

    In fact, the following uniqueness result holds.

    Theorem: If G and G’ are subgroups of S12 isomorphic to M12 then they are conjugate in S12.

    (This may be found in [7], pg 211.)

    6. Presentations

     

    The presentation of M12 will be shown later, but first I will define a presentation.

    Let G = < x1,…,xn | R1(x1,…xn) = 1, …, Rm(x1,…,xn) = 1> be the smallest group generated by x1,…,xn satisfying the relations R1,…Rm. Then we say G has presentation with generators x1,…,xn and relations R1(x1,…xn) = 1, …, Rm(x1,…,xn) = 1.

    Example: Let a = (1,2,…,n), so a is an n-cycle. Let Cn be the cyclic group, Cn = < a > =
    {1,a,…,an-1}. Then Cn has presentation < x | xn=1 > = all words in x, where x satisfies xn.=1 In fact, < x | xn = 1 > is isomorphic to < a >. Indeed, the isomorphism
    < x | xn = 1 > –> < a > is denoted by xk –> ak, 0 <= k <= n-1. Two things are needed for a presentation:

    • generators, in this case x, and
    • relations, in this case xn = 1.

    Example: Let G be a group generated by a,b with the following relations; a2 = 1, b2 = 1, (ab)2 = 1:

    G = < a,b | a2 = 1, b2 = 1, (ab)2 = 1 > = {1,a,b,ab}.

    This is a non-cyclic group of order 4.

    Two presentations of M12 are as follows:

    M12 = < A,B,C,D | A11 = B5 = C2 = D2 = (BC)2 = (BD)2 = (AC)= (AD)3 = (DCB)2 = 1, AB =A3 >

    = < A,C,D | A11 = C2 = D2 = (AC)3 = (AD)3 = (CD)10 = 1, A2(CD)2A = (CD)2 >.

    In the first presentation above, AB = B-1AB. These are found in [6] and Chap. 10 Sec. 1.6 [12].

    7. Crossing the Rubicon

     

    The Rubicon is the nick-name for the Rubik icosahedron, made by slicing the icosahedron in half for each pair of antipodal vertices. Each vertex can be rotated by 2*pi/5 radians, affecting the vertices in that half of the Rubicon, creating a shape with 12 vertices, and six slices. The Rubicon and M12 are closely related by specific moves on the Rubicon.

    Let f1, f2, …,f12 denote the basic moves of the Rubicon, or a 2*pi/5 radians turn of the sub-pentagon about each vertex. Then according to Conway,

    M12 = < x*y-1 | x,y are elements of {f1, f2, …,f12 } >.

    Actually, if a twist-untwist move, x*y-1, as above, is called a cross of the Rubicon, then M12 is generated by the crosses of the Rubicon! ([1], Chap. 11 Sec. 19 of [12])

    8. M12 and the Minimog

     

    Using the Minimog and C4 (defined below), I want to construct the Golay code GC12.

    The tetracode C4 consists of 9 words over F3:

      0 000,     0 +++,    0 ---,         where 0=0, +=1, and -=2 all mod 3.
      + 0+-,     + +-0,    + -0+,
      - 0-+,     - +0-,    - -+0.
    

    Each (a,b,c,d) in C4 defines a linear function f : F3 –> F3, where f(x) = ax+b, f(0) = b, f(1) = f(+) = c, f(2) = f(-) = d, and a is the “slope” of f. This implies a + b = c (mod 3), b – a = d (mod 3).

    Minimog: A 4×3 array whose rows are labeled 0,+,-, that construct the Golay code in such a way that both signed and unsigned hexads are easily recognized.

    A col is a word of length 12, weight 3 with a “+” in all entries of any one column and a “0” everywhere else. A tet is a word of length 12, weight 4 who has “+” entries in a pattern such that the row names form a tetracode word, and “0” entires elsewhere. For example,

     
                 _________          _________
                 | |+    |          | |+    |
                 | |+    |          |+|  +  |
                 | |+    |          | |    +| 
                 ---------          ---------               
                 "col"              "tet"
    

    The above “col” has “+” entries in all entries of column 2, and “0” entries elsewhere.
    The above “tet” has a “+” entry in each column. The row names of each “+” entry are +, 1, +, – respectively. When put together, + 0+- is one of the nine tetracode words.

    Lemma: Each word belongs to the ternary Golay Code GC12 if and only if

    • sum of each column = -(sum row0)
    • row+ – row is one of the tetracode words.

    This may be found in [4].

    Example:

    |+|+ + +|      col sums: ----      row+ - row-: --+0
    |0|0 + -|      row0 sum: + = -(sum of each col)
    |+|+ 0 -|
    
    

    How do I construct a Golay code word using cols and tets? By the Lemma above, there are four such combinations of cols and tets that are Golay code words. These are: col – col, col + tet, tet – tet, col + col – tet.

    Example:

      col-col         col+tet      tet-tet       col+col-tet
    
     | |+   -|       | |+ +  |    |+|0 + +|      | |- + +|
     | |+   -|       |+|  -  |    |-|  -  |      |-|  0 +|
     | |+   -|       | |  + +|    | |    -|      | |  + 0| 
      ? ? ? ?         + 0 ? -      - ? - +        + 0 + -
    

    “Odd-Man-Out”: The rows are labeled 0,+,-, resp.. If there is only one entry in a column then the label of the corresponding row is the Odd Man Out. (The name of the odd man out is that of the corresponding row.) If there is no entry or more than one entry in the column then the odd man out is denoted by “?”.

    For example, in the arrays above, the Odd-Men-Out are written below the individual arrays.

    For the Steiner system S(5,6,12), the minimog is labeled as such:

                                  ______________
                                  |0  3 inf  2 |
                                  |5  9  8  10 |
                                  |4  1  6  7  |
                                  --------------
    

    The four combinations of cols and tets above that construct a Golay code word yield all signed hexads. From these signed hexads, if you ignore the sign, there are 132 hexads of the Steiner system S(5,6,12) using the (o, inf, 1) labeling discussed in Section 9 below. There are a total of 265 words of this form, but there are 729 Golay code words total. So, although the above combinations yield all signed hexads, they do not yield all hexads of the Golay code ([12] pg. 321).

    The hexad for the tet-tet according to the S(5,6,12) Minimog above would be (0, inf, 2, 5, 8, 7).

    The rules to obtain each hexad in this Steiner system is discussed in Section 9 below.

    A Steiner system of type (5,6,12) and the Conway-Curtis notation can be obtained from the Minimog. S12 sends the 3×4 minimog array to another 3×4 array. The group M12 is a subgroup of S12 which sends the Minimog array to another array also yielding S(5,6,12) in Conway-Curtis notation.

    9. Kitten

     

    The kitten is also an interesting facet of the Minimog. Created by R.T. Curtis,
    kittens come from the construction of the Miracle Octal Generator, or MOG, also created by R.T. Curtis. (A description of the MOG would be too far afield for this post, but further information on the MOG can be gotten from [3] or [6].)

    Suppose we want to construct a Steiner system from the set T = {0, 1, …, 10, inf}.
    The kitten places 0, 1, and inf at the corners of a triangle, and then creates a rotational symmetry of triples inside the triangle according to R(y) = 1/(1-y) (as in [2], section 3.1). A kitten looks like:

                                    infinity
    
                                       6
    
                                    2     10
    
                                 5     7      3
    
                              6     9      4     6
    
                           2    10     8      2     10
    
                     0                                    1
    
                                Curtis' kitten               
    

    where 0, 1, inf are the points at infinity.

    Another kitten, used to construct a Steiner system from the set T = {0, 1, …, 10, 11} is

                                       6
    
                                       9
    
                                    10     8
    
                                 7     2      5
    
                              9     4     11     9
    
                          10     8     3      10     8
    
                     1                                    0
    
                             Conway-Curtis' kitten
    

    The corresponding minimog is

                      _________________________
                      |  6  |  3  |  0  |  9  |
                      |-----|-----|-----|-----|
                      |  5  |  2  |  7  | 10  |
                      |-----|-----|-----|-----|
                      |  4  |  1  |  8  | 11  |
                      |_____|_____|_____|_____|
    

    (see Conway [3]).

    The first kitten shown consists of the three points at 0, inf, 1 with an arrangement of points of the plane corresponding to each of them. This correspondence is:

             6 |10 | 3              5 | 7 |3               5 | 7 | 3 
             2 | 7 | 4              6 | 9 |4               9 | 4 | 6 
             5 | 9 | 8              2 |10 |8               8 | 2 |10
     
            inf-picture             0-picture              1-picture
    

    A union of two perpendicular lines is called a cross. There are 18 crosses of the kitten:

                    ___________________________________________
                    |* * * |* * * |* * * |*     |  *   |    * |
                    |*     |  *   |    * |* * * |* * * |* * * |
                    |*     |  *   |    * |*     |  *   |    * |
                     -----------------------------------------
                     _________________________________________
                    |*     |  *   |* *   |*     |*   * |    * |
                    |*     |  *   |* *   |  * * |  *   |    * |
                    |* * * |* * * |    * |  * * |*   * |* * * |
                     -----------------------------------------
                     _________________________________________
                    |*   * |    * |  * * |  *   |  * * |* *   |
                    |*   * |* *   |*     |*   * |  * * |    * |
                    |  *   |* *   |  * * |*   * |*     |* *   |
                    ------------------------------------------
    
    

    A square is a complement of a cross. The 18 squares of a kitten are:

                    ___________________________________________
                    |      |      |      |  * * |*   * |* *  |
                    |  * * |*   * |* *   |      |      |     |
                    |  * * |*   * |* *   |* *   |*   * |* *  |
                     -----------------------------------------
                     _________________________________________
                    |  * * |*   * |    * |  * * |  *   |* *   |
                    |  * * |*   * |    * |*     | *  * |* *   |
                    |      |      |* *   |*     |  *   |      |
                     -----------------------------------------
                     _________________________________________
                    |  *   |* *   |*     |*   * |*     |    * |
                    |  *   |    * |  * * |  *   |*     |* *   |
                    |*   * |    * |*     |  *   |  * * |    * |
                     -----------------------------------------
    

    The rules to obtain a hexad in the {0,1,inf} notation are the following:

    • A union of parallel lines in any picture,
    • {0, 1, inf} union any line,
    • {Two points at infinity} union {square in a picture corresponding to omitted point at infinity},
    • {One point at infinity} union {cross in the corresponding picture at infinity}.

    (See [2])

    M12 is isomorphic to the group of automorphisms of the Steiner system S(5,6,12) in the Conway-Curtis notation.

    10. Mathematical Blackjack or Mathieu’s 21

    Mathematical Blackjack is a card game where six cards from the group {0,1,…,11} are laid out face up on a table. The rules are:

    • each player must swap a card with a card from the remaining six, that is lower than the card on the table;
    • the first player to make the sum of all six cards less than 21 loses.

    According to Conway and Ryba [8, section V, part (d)], the winning strategy of this game is to choose a move which leaves a Steiner hexad from S(5,6,12) in the shuffle
    notation, whose sum is greater than or equal to 21, on the table.

    The shuffle notation for the hexad, used in the Mathematical Blackjack game, is shown below (see also the description in the hexad/blackjack page):

                  8 |10 |3            5 |11 |3            5 |11 |3
                  9 |11 |4            2 | 4 |8            8 | 2 |4 
                  5 | 2 |7            7 | 9 |10           9 |10 |7 
             
                 0-picture          1-picture          6-picture
    

    Riddle: Assuming the strategy, player A just made a winning hexad move that will force player B to make the sum under 21 on his next turn. Joe Smith walks up to player B and offers to shuffle all 12 cards while player A isn’t looking, for a fee. Player B grabs at his chance thinking that a random shuffle will let him back in the game. How is it that player B still loses?

    Joe is actually working for Player A. Joe does not shuffle the cards randomly, but instead uses the M12 group generated by r, s (see section 1) to shuffle the cards. Since the M12 group preserves hexads, player A still has a winning game. (He and Joe split the money.)

    11. Sporadic Groups

     

    A simple group is a group with no normal subgroups except itself and {1}. Most simple groups are from a family such as PSL2(Fp), p>3 or An, n >= 5. However there exist some simple groups outside of such well known families. These are called sporadic simple groups. M12 is a sporadic simple group of order 95,040. The only smaller sporadic group is M11 of order 7,920. (See [10] pg. 211)

    12. Stabilizer in M24 of a dodecad.

     

    M24 is a sporadic simple group of order 244,823,040 containing M12 as a subgroup. The Steiner system S(5,8,24) is a collection of 8 element subsets, called octads, from a 24 element set X, with the property that any five elements in X determine a unique octad in the system. There are (24 choose 5)/(8 choose 5) = 759 of these octads. M24 is the subgroup of SX which sends the set of octads to itself. Two octads, O1, O2, intersect in a subset of X of order 0,2,4,6 or 8 [14]. If |O1 intersect O2| = 2 then O1 + O2 is order 12. Such a subset of X is called a dodecad. M12 is isomorphic to

    {g in M24 | g(O1 + O2) = (O1 + O2)} = the stablizer of the dodecad O1 + O2.
    (See [6] for details)

    Much more information can be received from the references below or from the hexad/blackjack page.

    References

     

    1. W. D. Joyner, Mathematics of the Rubik’s Cube (USNA Course notes), 1997.
    2. R. T. Curtis, “The Steiner System S(5,6,12), the Mathieu Group M12 and the ‘Kitten’ ,” Computational Group Theory, Academic Press, London, 1984.
    3. J. H. Conway, “hexacode and Tetracode- MOG and MINIMOG,” Computational Group Theory (ed. Atkinson), Academic Press, London, 1984.
    4. Vera Pless, “Decoding the Golay Code,” Transactions of Information Theory, IEEE, 1986, (pgs 561-567).
    5. R. T. Curtis, “A new Combinatorial approach to M24“, Mathematical Proceeding of the Cambridge Philosophical Society, Vol. 79, 1974.
    6. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, “M12,”,
      Atlas of Finite Groups, Clarendon Press, Oxford, 1985.
    7. Robinson, A Course in the Theory of Groups, Springer, 1996.
    8. J. H. Conway, N. Sloane, “Lexicographic Codes: Error-Correcting Codes
      from Game Theory,” Transactions on Information Theory, IEEE, 1986.
    9. A .Adler, “The modular Curve X(11) and the Mathieu group M11“,
      Proc. London Math Society 74(1997)1-28.
      See also the paper X(11) and M11.
    10. T. Thompson, From Error-Correcting Codes Through Sphere
      Packings to Simple Groups
      , The Mathematical Association of
      America, 1983.
    11. Rotman, J, Introduction to the Theory of Groups, 4th ed.
      Springer-Verlag, 1995.
    12. J. Conway, N. Sloane, Sphere Packings, Lattices, and Groups,
      Springer-Verlag, 3rd ed., 1999.
    13. B. Kostant, “The Graph of the truncated icosahedron and the
      last letter of Galois.” Notices of the A.M.S. 42(1995)959-
      968.
    14. E. Assmus, “On the Automorphism Groups of Paley-Hadamard
      Matrices.” Combinatorial Mathematics and its Applications.
      University of North Carolina Press, 1969, (pgs 98-103).
    15. P. Greenberg, Mathieu Groups, Courant Institute of Math and
      Science, New York University, 1973.
    16. P. Cameron, J. Van Lint, Designs, Graphs, Codes, and Their
      Links
      , London Mathematical Society, Cambridge University
      Press, 1991.
    17. F. MacWilliams, N. Sloane, The Theory of Error Correcting
      Codes
      , North Holland Publishing Company, 1978.
    18. R. Graham, P. Diaconis, W. Kantor, “The Mathematics of
      Perfect Shuffles”, Advanced Applied Math, Vol. 4, 1985, (pgs
      175-196).

    Typed into html by wdj, 4-18-97.
    Corrections 4-27-2001.
    Last updated 2018-06-10.