User:Watchduck/hat
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.
Contents
Boolean functions[edit]
 For equivalence classes like secs and becs see Equivalence classes of Boolean functions.
Nibble shorthands[edit]
For some purposes I use a set of selfdeveloped 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]
When finite subsets are to be ordered in a sequence, it is often better to order them like reflected binary numbers (littleendian)  although for most people ordering them like binary numbers would be more intuitive.
The subsets of {A,B}

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


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

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...}.
examples  


A more general concept is colexicographic order (see lexicographic and colexicographic order).
Dual matrix[edit]
When a matrix A is an m×n matrix, containing p×q matrices B_{ij} as elements,
it is often interesting to see the dual matrix X, which is a p×q matrix, containing m×n matrices Y_{ij} 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 b_{ij,kl} in the matrix B_{ij}
is the same as
the element y_{kl,ij} in the matrix Y_{kl} .
The matrix
is dual to
.
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 6bit binary vectors,
but a free arrangement was chosen, to serve symmetry.
Join table 

(Compare: Symmetric_group_S4#Join_and_meet) 
Gould'sMorse 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 ( A001316),
and the minus signs are distributed like the ones in Thue–Morse sequence ( A010060).
So it is the Hadamard product of the Gould's sequence and A106400.
See: File:LDU decomposition of Walsh 16.svg, File:Walsh 16 * Gould'sMorse.svg, File:Walsh 8 * Gould'sMorse.svg