User:Mate2code/hat

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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 doesn't exist, and have to choose one on my own.



Boolean functions[edit]

For equivalence classes like secs and becs see Equivalence classes of Boolean functions.

Nibble shorthands[edit]

commons:Category:Nibble shorthands
Tesseract Hasse diagram
with nibble shorthands

For some purposes I use a set of self-developed symbols for the 16 binary strings with 4 digits (nibbles).
They just represent the strings themselves, and not anything they may stand for.

They may be assigned to numbers from 0 to 15 like binary or like reverse binary numbers, I usually do the latter.


Reverse binary[edit]

Reverse binary warning sign

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[edit]

16×16 matrix of 1×4 matrices

Below the dual 1×4 matrix
of 16×16 matrices
4x16 matrix of 1x4 matrices

Below the dual 1x4 matrix
of 4x16 matrices

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.







Gould's-Morse sequence[edit]

This portmanteau name I use for the following sequence, which occurs when dealing with Walsh matrices:
1, -2, -2, 4,    -2, 4, 4, -8,    -2, 4, 4, -8,    4, -8, -8, 16,
-2, 4, 4, -8,    4, -8, -8, 16,    4, -8, -8, 16,    -8, 16, 16, -32 ...

The absolute values are from Gould's sequence (OEISA001316),
and the minus signs are distributed like the ones in Thue–Morse sequence (OEISA010060).
So it is the Hadamard product of the Gould's sequence and OEISA106400.

See: File:LDU decomposition of Walsh 16.svg, File:Walsh 16 * Gould's-Morse.svg, File:Walsh 8 * Gould's-Morse.svg