New Ideas.

To achieve more characterizations of designs, create classes of designs based on constructions, maybe topology or poset.

Look at Music. If we look at a song, I feel from my understanding it is a parameterized surface, such that each time value outputs different magnitudes of an array of Hz. Then a note would be just a subsurface. Then looking at patterns of repeated subsurfaces could maybe reveal something.

This is it.

I found it. Kreher wrote a moebius demystified. I need to read it, understand it, then go to SAGE and see what is implemented. If it's not, do it myself.

Topics of interest:

• Subgroup Lattice
• Subgroups(as posets)
• Some how the orbit counting lemma is involved
• Applying the moebius inversion
• Generalized $A_{t,k}$ matrices
• (Covering Arrays in the Background)

Basic Math.

Let $A,B,C \in \{0,1,\infty,\alpha\}$, be such that $g(\{A,B,C\})=\{0,1,\infty\}$. Define
\[
h(x)=h_{A,B,C}(x)=\frac{(x-A)(B-C)}{(x-C)(B-A)}.
\]
Then $h(\{A,B,C\})=\frac{1}{h}(\{A,B,C\})=\{0,1,\infty\}$.

My question is as follows: In any mapping, at all, if you have three elements mapping to the same thing those three things map to in the inverse, then all those three elements are the same, and then $\alpha$ cannot be apart of $A$, $B$, or $C$. amirite?

I'm still sick something awful. but this little bit tripped me up in the paper I'm reading, and I can't follow the proof without understanding this point.

Permutation Groups. Automata. Presentations.

I'm in that stage of being interested in something where I want to know waaaaaay more than I know right now. First, look at this problem from Peter Cameron's collection of problems:

Problem 79
The following problem in design theory comes (indirectly) from automata theory.

Let $m$ be an integer greater than 3. Does there exist a $t$-$(2m,m,λ)$ design, for some $t>1$ and some $λ>0$, with the properties

the complement of a block is a block,
the number of blocks properly divides the total number of m-subsets of the point set?
The Hadamard conjecture would imply that the answer is "yes" for all $m>3$; but perhaps there is an easier construction, since $t$ and $λ$ are not specified.

Like, take a look at this pdf.
I emailed Cameron, and he said he's writing a "substantial paper" about the links between permutation groups and Automata. Now, Automata was my first love; the hours I spent on Formal Models homework was the most fun I've had doing homework. But, I know NOTHING yet about permutation groups. I mean, sure I've had classes, but much that I read in Cameron's Permutation Groups book is new to me.

So I end the blog post with a plea to find out what this set is composed of. $G$ is a group, $A$, and $B$ are subgroups, while $\phi$ is an isomomorphism between $A$ and $B$.
\[
\langle G, t : t^{-1}at=a\phi \text{ for all }a \in A\rangle
\]
It is larger than $G$, so it could be some sort of product.

I looked up HNN on wiki, and it is supposed to be a presentation. I probably just missed where he defined his notation about presentations. Oh well, back to thinking small; I need to study more. I'm quite quite sick today; my whole body aches something fierce.

Google AI Challange.

I am going to try to participate in the Google AI Contest, writing a Tron AI. Wish me luck. From the videos, there seem to be two points in the game:

1. (Online) Trap Them / Isolate yourself.

2. (Online) Find a maximally Hamiltonian path on your isolated sub graph.

Current Ideas:

(First some preliminaries, I vision the field as a Graph, with a vertex set and edge set.)
(Concerning topic 1.)
Find min-cuts in the graph. All of them within a certain criteria.
1. If the graph is more open (larger min-cuts, with little below max) then an openly aggressive opponent strategy would seem appropriate.
2. If the graph is more closed (very low min-cuts, with several near minimal) then a shortest path to cut off more than 50% of the map would seem more appropriate.

(Concerning topic 2.)
As the game no longer depends on what the opponent does, calculate maximally covering(with exits!) successively bigger in size.

That's all for now. Off to develop topic 1.

Ich Bin Ein Gruppenpest.

So I was doing some basic Combinatorics tonight, working on a Cameron problem, and I realized I utterly fail at it.

Like, here is a skill I feel I should know by know: Given an arbitrary expression $E$ in terms of a few variables, come up with a combinatorial problem such that the $E$ is the solution.

Example.

Set $E$ to be $m!$. The answer could then be any number of things, with an acceptable answer being "How many permutations of $m$ points are there?"

However, I have something a bit more problematic at hand, and it is really close to something simple, but not quite there. I feel frustrated.

Things I would like to know about Design Theory.

I would like to know:

• All ways to get new designs from old.
• All theorems concerning automorphisms of designs.
• All Combinatorial Representation Theory.
• All Combinatorial group invariant theory.