User:Watchduck/hat

Habits and terminology
I try to use technical terms that are generally accepted, but sometimes I don't know a common name,
possibly because it does not exist, and have to choose one on my own.

Boolean functions

For equivalence classes like SECs and BECs see Equivalence classes of Boolean functions.

Nibble shorthands

commons:Category:Nibble shorthands

For some purposes I use a set of self-developed signs for the 16 binary strings with 4 digits (nibbles).

Their horizontal symmetry is like that of the nibbles themselves. The rotation of a sign belongs to the complementary nibble.
(Hence the vertical reflection gives the reflected complement, which is e.g. the relationship between AND and OR).
The signs of asymmetric nibbles have their weight on the same side as the nibble (i.e. the side with more 1s).

The signs represent the nibbles themselves, and not anything they stand for.
But the similarity of and to 3 and 7 makes the big-endian interpretation (as hexadecimal figures) more intuitive than little-endian. Representing tesseract coordinates and subsets of a 4-element set. Technically without numerical interpretation, but rather little-endian through the connection to this file. Representing numbers 0...15. Clearly big-endian.

Reverse binary

When finite subsets are to be ordered in a sequence, it is often better to order them like reflected binary numbers (little-endian) - although for most people ordering them like binary numbers would be more intuitive.

The subsets of {A,B}
ordered like binary numbers are:

 { } { B } { A } { A, B }
The subsets of {A,B,C}
ordered like reflected binary numbers are:
 { } { A } { B } { A, B } { C } { A, C } { B, C } { A, B, C }
The subsets of {A,B,C}
ordered like binary numbers are:
 { } { C } { B } { B, C } { A } { A, C } { A, B } { A, B, C }

Only when the subsets are ordered like reflected binary numbers, the sequence of subsets of {A,B}
is the beginning of the sequence of subsets of {A,B,C}.
This leads to a sequence of finite subsets of the infinite set {A,B,C,D...}.

A more general concept is colexicographic order (see lexicographic and colexicographic order).

Dual matrix

When a matrix A is an m×n matrix, containing p×q matrices Bij as elements,
it is often interesting to see the dual matrix X, which is a p×q matrix, containing m×n matrices Yij as elements.

Dual matrices contain the same elements of elements (usually that should be numbers),
so in the end they show the same information, but in a different way.

The element bij,kl in the matrix Bij
is the same as
the element ykl,ij in the matrix Ykl .

The matrix
$A={\begin{pmatrix}B_{11}&B_{12}&B_{13}\\B_{21}&B_{22}&B_{23}\end{pmatrix}}={\begin{pmatrix}{\begin{pmatrix}a&b\\c&d\end{pmatrix}}&{\begin{pmatrix}1&2\\3&4\end{pmatrix}}&{\begin{pmatrix}\alpha &\beta \\\gamma &\delta \end{pmatrix}}\\{\begin{pmatrix}e&f\\g&h\end{pmatrix}}&{\begin{pmatrix}5&6\\7&8\end{pmatrix}}&{\begin{pmatrix}\epsilon &\zeta \\\eta &\theta \end{pmatrix}}\end{pmatrix}}$ is dual to

$X={\begin{pmatrix}Y_{11}&Y_{12}\\Y_{21}&Y_{22}\end{pmatrix}}={\begin{pmatrix}{\begin{pmatrix}a&1&\alpha \\e&5&\epsilon \end{pmatrix}}&{\begin{pmatrix}b&2&\beta \\f&6&\zeta \end{pmatrix}}\\{\begin{pmatrix}c&3&\gamma \\g&7&\eta \end{pmatrix}}&{\begin{pmatrix}d&4&\delta \\h&8&\theta \end{pmatrix}}\end{pmatrix}}$ .

$b_{11,11}=a=y_{11,11}~$ $b_{23,21}=\eta =y_{21,23}~$ This concept is not limited to matrices, as the following example shows.
The join table is a 24×24 matrix. The inversion sets could be displayed by 6-bit binary vectors,
but a free arrangement was chosen, to serve symmetry.