Help a newb out with Design Theory

I have a design $D=(X,B)$ with parameters $t$-$(v,k,\lambda)$.

Let $P^*=(P,\subset)$ be the poset (P is also an abstract simplicial complex)
formed by the following construction:

1. $X \in P$.
2. $B$ is a subset of $P$. (call these facets)
3. If $Y$ is a subset of $b \in B$, then $Y$ is in $P$.

With poset relation subset.

Obviously, all $t$ subsets of $X$ are in this poset, and they have $\lambda$
facets (blocks) that contain them.

Define a new parameter $c$.

Let$ t\leq c\leq k$.
There is a smallest $i$ such that all $i$ subsets of $X$ are in $P$,
but not all $i+1$ subsets are in $P$. Call this smallest integer $c$.

For complete designs, $c$ is $k$.

Does this parameter have a name? does it have a use?

Without loss we can say $t$ is the greatest such $t$ for this design. If $twhat can we say about the number of facets(blocks) that contain these $c$ subsets? nothing?

First recursive function I've written in years.

recurse:=function(G,B,i,g,f)
  local j;
  if Size(g) = (i-1) then
    if Size(g) = Size(B) then
      return f(g);
    else
      for j in Combinations(G,B[i]-1) do
        return recurse(G,B,i+1,Concatenation(g,[j]),f);
      od;
    fi;
  else return;
  fi;
end;


global_array:=[];
fun:=function(G,B,g)
  local gset,glist,difflist,gcopy,x,L,i,j,pos,idealdifflist;
#TODO Implement FindL, code's written, just wrap it up
  L:=2;#FindL(G,B);
  gcopy:=StructuralCopy(g);
  gset:=Flat(gcopy);
  glist:=List(G);
  difflist:=ConstantArray(0,Size(G));
  idealdifflist:=ConstantArray(L,Size(G));
  idealdifflist[Position(glist,One(G))]:=0;
  # if Id is not any one of the diffs;
  if Positions(gset,One(G))<>[] then
    return;
  else
    for i in [1..Size(gcopy)] do
      Add(gcopy[i],One(G));
    od;
#generate difflist
    for i in [1..Size(gcopy)] do
      for j in Tuples(gcopy[i],2) do
        pos:=Position(glist,j[1]*j[2]^-1);
        difflist[pos]:=difflist[pos]+1;
      od;
    od;
  fi;
#verify difflist
  if difflist = idealdifflist then
    Add(global_array,g);
  fi;
end;

Splines (bill theory of OCR)

To preface this, I have only read wiki articles, and have barely skimmed them at that.

Given an experimentally found spline $S_1$ and a two dimensional spline array $SA$, we want to find the "closest fit" spline for $S_1$. The structure of $SA$ is such that each column $c$ has multiple forms of a given spline $S_c$.

Alright, and I want to apply BCH codes to do this. I'm going to take coding theory next semester, so I guess I'll just wait until then to think about this more.

Edit 1: I just came up with two ideas while laying in bed, among others coming to me right now.

1. measure correspondence of overlapping segments of splines.
2. Create a 3 dimensional graph with the z being the first derivative of the spline, another with the z being the 2nd derivative, and another with the z being the 3rd derivative.
3. Assume there are building blocks of shapes, break each spline into it's fundamental composition of shapes.
4. the input is also not just one spline, but could be several different spline interpretations.

Edit 2: Syndrome's apparently is what I need. I'll make sure to remember that when I take the class.

Edit 3(March 13, 2010): Computer Vision is a pretty large field, and this is a very mathematical approach to it, so I think I should read and play with these things before I see where this approach fits into this quite large field.

Numerical Geometry of non-rigid shapes

Numerical Geometry of Images

Learning OpenCV

NURBS

Gestalt

Abstract vs. Applied

So I'm working on two projects currently, one is extremely applied, and hopefully will be able to generate a general theory from this experimentation. The other is working on a very general theory that someone has already demonstrated to be useful. It's sort of weird. I love the general project WAY more, even though the first applied project is breaking new ground. I'm not sure what to make of that.

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.