Posts : 95 Join date : 2009-09-15 Age : 57 Location : Alexandria

Subject: Re: Encyclopedia of Integer Sequences Wed Nov 17, 2010 3:37 pm

Yes I have. Take N×N with (a, b) + (c, d) = (a+b, c+d) and (a, b).(c, d) = (ad + bc +ac, ad + bc + bd) ; then 1 = (1, 0) 1 = (0, 1).

There is nothing unusual here...

The arity of X is (1,0) and the arity of X is (0,1). Tracking back,

The arity of F_{n} is (n, 0) and the arity of 1/F_{n} is (0, n).

Example : The Wonder Species Pascal , the natural generalization of Ens. I have a set-box , but only a part of the slots are visible in a mirror. I stick a cube. The stuck cube is either visible in the mirror, or not.

Subject: Re: Encyclopedia of Integer Sequences Wed Nov 17, 2010 8:35 pm

nick wrote:

The Wonder Species Pascal , the natural generalization of Ens. I have a set-box , but only a part of the slots are visible in a mirror. I stick a cube. The stuck cube is either visible in the mirror, or not.

Pascal' = Pascal'^{R} + Pascal'^{M}

Bro ? do you believe that ? Pascal' = Pascal'^{R} + Pascal'^{M} = Pascal

Anyway Nick, good work ! Here is a dedication to you :

Posts : 95 Join date : 2009-09-15 Age : 57 Location : Alexandria

Subject: Re: Encyclopedia of Integer Sequences Fri Nov 19, 2010 1:28 pm

G funk wrote:

Transposition = X.X = X + X either you put an egg in the mirror cup, or you put it in the real cup. Anyway, after placing one egg, the other cup is automatically fulled.

Par Lewis Carroll ! I will maintain the name of Wonder Species ! Here is how it works :

X(X) = X(X)= X.X = X + X

I have only one cube; I place it in some mirrored real slot or in a real mirrored slot. In both cases, I obtain a filled slot in the Real world and one in the Mirror world. The Wonder Table is :

The word "arity" stands for the wonder cardinality (n, m). (the old "arity" became useless by passing to species, that are independent of arity)

And here starts a new inquire, - what known things fit to the Arithmetics of Arities and - they get a new shorter formulation using the four basic operations plus the transposition :

1/F or, equivalently, F ?

========= Let me note a good news, At CERN have trapped 38 anti-atoms, atoms made of anti-matter.

The above suggests the notation } a { for an anti-individual.

Posts : 95 Join date : 2009-09-15 Age : 57 Location : Alexandria

Subject: Re: Encyclopedia of Integer Sequences Sat Dec 18, 2010 11:13 pm

I think some more explanations are needed, even at Math Club was already explained the difference between combinatorial groups and topological groups.

There is a logical explanation of the combinatorial fact that Groupe'=Lin.

As known, one of the secrets of the combinatorial identities is that they catch an object and its very next context.

Hence, Elem = X.Ens reflects the very next context of an element (X) inside a set contains the other elements forming a set (Ens). Subset = Ens.Ens says that the very next context of a subset (the first Ens) contains also the complementary subset (the second Ens).

Similarly, the very next context of a group contains the other permutations of the Symmetric group. A group of order n is also a transitive (Burnside) subgroup of Sym(n) (Cayley).

We have a one-to-one correspondence between - groups - combinatorial groups - transitive subgroups of Sym(n) (of order n) - actions of Sym on the cosets of the target group - molecular species Groupe.

While deriving at the fourth hyphen level, one gets a Lin. (And I am also working to the Cubes-and-Puzzles explanation)

Subject: Re: Encyclopedia of Integer Sequences Sun Dec 19, 2010 12:45 am

Nick, sometimes the most obvious things are the hardest to explain.

Either you take your time and you draw an extended Cayley table, of n×n!, instead of a regular n×n table,

Or you take your time and explain with caution the sticking theory, because no one likes stuck pieces. In the real world, stuck pieces means broken mechanisms.

Both cases, it is a nice winter here... let's enjoy it !