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 Encyclopedia of Integer Sequences

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nick
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PostSubject: Re: Encyclopedia of Integer Sequences   Mon Nov 01, 2010 9:18 pm

You know, doc,

I am thinking to another T-shirt, that would express my postulate.

How to express, in a short formula, that the identity is an even permutation ? that the signature (discriminator) is well defined ? that I cannot solve all puzzles, unless they means to put objects in the same bag ?

How about A' = A ?

Anyway, it will imply that 1+ 1==2, meaning that there are two realizations for the Altern Species, for each n, the same amount as I could have a choice to put n objects or in a bag, either in another bag.
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PostSubject: Re: Encyclopedia of Integer Sequences   Mon Nov 01, 2010 10:14 pm

nick wrote:
another T-shirt,
Bro, this fellow does not have a problem with the G-word, but with his T-shirts.

Nick, you forgot the
N = L
T-shirt !

Par Peano and Zermelo, it looks nice!

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PostSubject: Re: Encyclopedia of Integer Sequences   Mon Nov 01, 2010 10:58 pm

Par Aristotle !

The formula is N = N = L and is reserved for my shirt.

The lateral N stands for the Named numberoids.
The middle N stands for the Ordinal Numbers, and
The L stands for the Lin species.

Obviously, you must be some kind of Aristotle disciples, since you have eliminated the middle term by the mnemonic rule of Aristotle for eliminating information while keeping some truth.

Let's clarify.

An axiom is something that must be taken for granted.
A postulate is something that must be taken for granted, but also it is obvious.

Are you capable to extract an axiom from my postulate ? By algebrisation ?



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PostSubject: Re: Encyclopedia of Integer Sequences   Tue Nov 02, 2010 12:01 am

nick wrote:
shirt.
Shirt ? what a shirt could be ?
Bro, maybe we should leave the T-shirt business and move to the shirt market !

Just imagine a shirt with ×({}) = 1 !


Par Peano and par Zermelo, Nick, you need an inductive definition of Sym(n), the group of bijections from an n-set to itself.
Then, instead of verifying all puzzles, you just verify the parity of the identity on Sym(2).
Then, you prove by complete induction that the parity of identity is even.

Believe me, no one will buy you shirt, that expresses just an incomplete induction.

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PostSubject: Re: Encyclopedia of Integer Sequences   Tue Nov 02, 2010 4:22 pm

Sorry, I do not see it.

I would obtain only some 1BUC = 1B. 1C

Nothing guarantees that the new 1BUC is not odd when 1B and 1C are even.

An equivalent sentence would be

A transposition is not a product of tri-cycles.

I think is the time to write some computer program, to proof this for large enough numbers. Maybe I will see an argument while programming.


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PostSubject: Re: Encyclopedia of Integer Sequences   Wed Nov 03, 2010 8:18 am

nick wrote:
A transposition is not a product of tri-cycles.

Par Fermat !

It looks like a conjecture.
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PostSubject: Re: Encyclopedia of Integer Sequences   Wed Nov 03, 2010 4:31 pm

So, do you think that I caught a conjuncturitis right in the mid-term ?

Doc, in the real world there are no transpositions.
If you want to swap two objects, you must have a third empty slot, let's say X.

Then, you move the object a from A to X,
Then, you move the object b from B to A,
Then, you move the object a from X to B.

One must make three operations to swap two objects. This is a Law of Nature doc. We live in an Altern World, doc.

A bag, or a set, or a set-box is a puzzle with two holes. Hence one can make all possible permutations.






Last edited by nick on Thu Nov 04, 2010 7:08 am; edited 1 time in total
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PostSubject: Re: Encyclopedia of Integer Sequences   Wed Nov 03, 2010 6:28 pm

Look, Nick, how it works :

Instead of taking as invariant the [parity (in Sym(16)) + taxicab distance] since the parity is not yet defined , do the following :

Rewrite your formulas collection with Up-Left, Up-Down etc. instead of Up, Left, Right, Down. It makes sense, since all your formulas have even length.

Now, take as invariant
[the taxicab distance for the hole plus the taxicab distances of the deranged cubes] modulo 4;

Each double-move does not affect this invariant. Moreover, if you compose two double-moves. you should have

Invariant (first double-move) + Invariant (second double-move) = Invariant (first.second) modulo 4;

Hence you need a 5x5 puzzle to verify this.

If anyone mess your puzzle by transposing two adjacent cubes, you cannot solve it, since the invariant is 2.


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PostSubject: Re: Encyclopedia of Integer Sequences   Wed Nov 03, 2010 7:19 pm

Nope. The counter-example is on 2x3. The discriminant should distinguish between any transposition and any cycle.

A transposition is a very abstract thing...

======== I have a good news ! Look =========

Suppose that identity has a double parity, both odd and even. Then, all permutations must have the same double parity, both odd and even.

A permutation P is a set of cycles. There is a parity delivered by this definition, the length of a shortest formula of decomposition in transpositions. Define thus the normal parity.

Suppose that permutations have abnormal decompositions.
Take a shortest abnormal decomposition for all permutations, let's say P= a.b.c....h.

Then aP has the other normal parity than P (to check*) while it has an abnormal decomposition b.c.....h, shorter than the shortest one, contradiction.

* the transposition either hits two distinct cycles, reuniting them - (and changing the normal parity with 1) or hits inside one single cycle, splitting it into two smaller cycles (and changing the normal parity with 1)

Fellows, the alternating group exists and it is different to symmetric group !


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PostSubject: Re: Encyclopedia of Integer Sequences   Thu Nov 04, 2010 10:24 am

nick wrote:
* the transposition either hits two distinct cycles, reuniting them - (and changing the normal parity with 1) or hits inside one single cycle, splitting it into two smaller cycles (and changing the normal parity with 1
Hey Nick, with such a switch you can burn a toy train !
A001563
n.n!

Switch = •Cyc + •Cyc.•Cyc
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PostSubject: Re: Encyclopedia of Integer Sequences   Thu Nov 04, 2010 1:51 pm

nick wrote:
I caught a conjuncturitis right in the mid-term ?
Definitively. The cause is your stereoscopic perception.

Take { a, b, c, d,...,n }.

One could say, : This is a set !
By stereoscopic perception, you see there : This is the set of the elements of a set.

A000027

ELEM = •ENS

You should denote the n realizations of the species with [ a, b, c,..., n ].

ELEM { a, b, c,..., n } = [a, b, c,...,n].


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PostSubject: Re: Encyclopedia of Integer Sequences   Fri Nov 05, 2010 10:16 pm

G funk wrote:
Switch = •Cyc + •Cyc.•Cyc

This switch seems to be a kind of operator... messing two cubes and transporting my box in the Wonderland, the other side (others sides) of the mirror...

(A+B)2 = A2 + B2
(A.B)2 = A2 .B + A.B2 + A.B

Given two boxes, A+B someone either messes tho cubes in A, or messes two cubes in B.
Given a box A.B, someone either messes tho cubes in A, either messes two cubes in B, or it points a cube in each box.

I need to obtain something like :
Cyc2= Cyc.Cyc A052517

Only one cycle will have electricity in the G funk's toy train. The other must be pushed manually.

I need some preliminaries.
1) Endospecies. An endospecies is a species where U=V=W is added to the definition. This delivers the definition by Sym(n) action on the set of realizations of an endospecies.
2) Cayley table of an action that contains in left side the table of Sym(n) and in right side the results B = sigma(A).
3) There are "transitive blocks" in a Cayley table, meaning derange-one-derange-all blocks. The first column of a transitive block contains all information and the others can be calculating using the first column and the rule of action.
4) The first column also contains the orbit-stab equation. Imagine that the first column is A, A, B, B, C, C. Stab(A) is a subgroup of Sym(3) and 2×3 = 6. Any proof must fit this remark.
5) Hence, there is a natural description of a transitive endospecies, as the action of Sym(n) on the cosets of Stab(A).
6) From this description one can extract a standardized description of a species, by forming listing the permutations of the cosets of Stab(A). For example,

Alt3 {a, b, c} = [{abc, bca, cab}, {bac, cba, acb}] = [A, B],
without having the possibility to specify here who is A and who is B. Anyway, since two species could be equipotent, they must have a potential, and this potential is the cardinality of [A, B], the collections of realizations.

I know that it is vulgarizing, but it could be useful to calculate the switched species by the technique of the Alternating Group Existence Deduction.

7) Now, a first remark is that abc can be encoded as {a, {a,b}, {b,c}} that means all transitive endospecies may be found, using the Set Science representation, among the nested sets of depth four:

Alt3 {a, b, c} = {{ {a, {a,b}, {b,c}}, bca, cab}, {bac, cba, acb}}.

This representation is immune to all possible permutations of the symbols a, b, c,...


Lin3 {a, b, c} = [{abc}, [{bca}, {cab}, {bac}, {cba}, {acb}] while
Perm3 {a, b, c} = [{abc, acb}, {abc, cba}, {abc, bac}, {abc, bca, cab}, {acb, cba, bac}, {abc, bca, cab, acb, cba, bac}]

Messing two cubes means to alter the notations. For each xyz...w I alter the notations interchanging for example, the first symbol with the second, obtaining yxz...w.

The result is a set (of depth four, just to humanize it) that has an inner structure, an endospecies defined on the same symbols.

------------------
He he, maybe you wonder why took me so long to "see the pussycat".
Fellows, three years ago I have tried to de-conceptualize the zero of a group (I was almost there sunny). Meanwhile, I had also some exams with plenty of zeros matrices. You should see my matrices exam paper flower

In reality, even the most abstract groups, which are the primitives of Lin, have a neutral element/zero/one/identity.

Par Rubik, every repeated formula brings will bring one day the cube in the same position.
For example, take some formulae F.F.F.F.F = G.G.G = H.H.H.H. These different writings stand for the same something, and that same something is the neutral element.

As endospecies (pure abstract groups), the 1 of a group is the 1 of the Sym(n).

------------
Another remark is that we can safely modify the first axiom of an equivalence relation see Wikipedia
from reflexivity to anti-reflexivity.
The trick is that elements must be either all related to themselves or all non-related to themselves.

When someone verifies this axiom, in fact he verifies that the elements are aligned to the same self-relation.
In practice, we may safely replace x==x with x <> x, where <> stands for the absence of relation with itself, but this will complicate the axioms and notations.

If a partition has at least a class with two elements, by applying the transitivity and symmetry to xRyRx we get xRx and all the elements self-identities must be polarized xRx instead of of non-xRx.
If a partition is the finest possible partition, with no two distinct elements related - it does not matter anymore if the elements are self-related or not.

This remark delivers :
As species, [{a, b, c, d, ... n}] and [{{a}, {b}, {c}, ...{n}}] are identical.
in the left side are non self-related elements, and in the right side are self related elements.

Ens= Ens(X)

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PostSubject: Re: Encyclopedia of Integer Sequences   Tue Nov 09, 2010 3:36 pm

nick wrote:
(A.B)2 = A2 .B + A.B2 + A.B
Bro, take a look at this proof :


Nick, you should try :

(F.G)2 = F2.G + F.G2 + X.F'.G'

where (X)2 = 0 and (X.X)2 = X

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PostSubject: Re: Encyclopedia of Integer Sequences   Tue Nov 09, 2010 5:37 pm

Quote :
Bro, take a look at this proof :
Nice one !

Nick, try this: Perm2 = X.Lin.Lin.Perm

Since Cyc2 = X.Lin.Lin, (the length of the toy-train is a full railroad) your deduction for the existence of the alternating group may be resumed :

In both cases, wherever the transposition hits, the local impact produces a X.Lin.Lin. Nevertheless, the remaining Perm is either odd or even. Odd perms plus even perms give perms.

Perm2 = Cyc2.Perm

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PostSubject: Re: Encyclopedia of Integer Sequences   Tue Nov 09, 2010 8:24 pm

G funk wrote:
Bro, take a look at this proof :
It's OK, bros.
You have a diagram that explains what unifying two slots in a puzzle box could mean. I have tried, and this one could be better :

The little 2 stands for the two cases : either unifying a slot from upper side with the lower one, or unifying a slot from the downside with an upside slot.

(F.G)² = F².G + F.G² + 2.X.F'.G' . Hence :

X²= 0 // no unification in a one-slot box
(X.X)²= 2.X // a two-slot box delivers two 1-slot box
Lin² = 2.X.Lin.Lin.Lin // by induction


... it seems that my new operation of unifying two slots is a simple F----> X.F''
Unifying two slots, I get a marked slot X while the rest of the puzzle has two sticked slots.
Here is the CCP :


Ens²= Elem // after unification in a bag, we got a non-empty bag with a special position.
Alt² = X.Alt // calculus
Cyc² = X.Lin.Lin // After unification, the unified slot became a special position
Perm² = 2.Cyc².Perm // the remaining perm after impact could be any perm.

and
F²' = F'² + F''
sticking a cube in an 2- unified box means : either I stick a non-unified slot, or I stick the unified one.
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PostSubject: Re: Encyclopedia of Integer Sequences   Thu Nov 11, 2010 6:59 pm

nick wrote:
If you want to swap two objects, you must have a third empty slot, let's say X.
Then, you move the object a from A to X,
Then, you move the object b from B to A,
Then, you move the object a from X to B.
One must make three operations to swap two objects. This is a Law of Nature doc. We live in an Altern World, doc.

Par Archimedes ! Doc I think I got the Abstract Transposition !

Doc, how about a distinction between out real wold, that contains no 2-cycles, and the world behind the mirror, the Wonderland ?

I mean, take the EGF of a cycle ln(1/(1+x)) ; then
2-cycles, 4 cycles, 6-cycles,... reside in the Wonderland, while
X, 3-cycles, 5-cycles, 7-cycles... belong to our real world ?

Anyway, every subgroup of Sym(n) is either made of even permutations (real-species) or made of an equal amount of odd and even permutations (wonder-species).

Thus :

Z2×Z2 (or P4bic) belongs to Our World, with e.g.f 6.x4

while

Cyc4 belongs to Wonderland, with e.g.f ­­–6.x4.

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PostSubject: Re: Encyclopedia of Integer Sequences   Fri Nov 12, 2010 10:05 pm

Nick.

the SGE/EGF engine works. It is up to you what you input in it. Nevertheless, since your deduction definitively established the existence of the Alternating Group, the Alternating Group should be someway caught in the algebraical framework.

Par Leibniz ! par Boole !

What they have delivered to you as some "introduction a la combinatoire" is some kind of algebrization of the Combinatorial Logic.

Not negotiable.

=================
I also have my problems :

Did the Bourbaky students had typewriters ? Were they IBM typewriters ?
Did Hilbert had typewriters ? Were they IBM typewriters ?


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PostSubject: Re: Encyclopedia of Integer Sequences   Sat Nov 13, 2010 2:01 am

Dr. Post wrote:
Not negotiable.
Ok Doc, I will screw my mind with the altern phenomenon logical comprehension.

Just let me note here six IBM expressions, that act by derangement on themselves :
x, 1-x, 1/x, x/(x-1), 1-1/x, and 1/(1-x)
They could mean the pass from wonder-species to complex-species.

============
Doc, this passing to alternating series would make sense only if it would bring a distinction between Z2×Z2 and Z4, or maybe between Perm and Lin. Otherwise, I got only a messy rewriting by duality.

It would be nice, a T-shirt GROUP' = ±LIN

============

Or could it be just about rotating the labels ?
...why should anyone rotate the labels/cubes/balls/eggs...

Here is the CCP :


Par Rubik ! it makes sense.
The vertexes of a cube can be rotated, to mess the cube.
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PostSubject: Re: Encyclopedia of Integer Sequences   Sun Nov 14, 2010 8:32 pm

Bro, did you took a look to the above CCP ?

Par Riemann ! we should type our CCPs on a sphere :

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PostSubject: Re: Encyclopedia of Integer Sequences   Sun Nov 14, 2010 10:19 pm

Here is the CGP.

After a 1/z transformation, an Abstract Complex Point gets on the other side of the screen.

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PostSubject: Re: Encyclopedia of Integer Sequences   Mon Nov 15, 2010 12:21 am

Par Democritus of Abdera !

Your abstract complex point looks like this, rather than an individual :

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PostSubject: Re: Encyclopedia of Integer Sequences   Mon Nov 15, 2010 1:13 am

Yes Doc, that is exactly what I was saying. An Abstract Complex Point is the thing that your hook and these old thumbtacks below have in common.

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PostSubject: Re: Encyclopedia of Integer Sequences   Mon Nov 15, 2010 1:15 pm

Bro, I think Nick is trying to say something like this :


Nick, you can't reach the other side of the screen with your compass.

Par Hamilton ! You should note the opposite points

1 ------> j
i -------> k

And so on. These special points acts on themselves by derangement; They must realize some kind of copy of your abstract complex point.

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PostSubject: Re: Encyclopedia of Integer Sequences   Tue Nov 16, 2010 1:19 pm

Bros, it's deeper than that. A normal subgroup is a group of group.

I must revise it from rational numbers fractions. Example : 4+1/3. We have :

4+1/3 = 4 + 3 (de-conceptualization)
4 + 3 = (4.3+1).3 (formula)
(4.3+1).3 = 13 3 (calculus)
13 3 = 13.(1 of 3) (formula)
13.(1 of 3) = (13/3).1 (definition)
(13/3).1 = 13/3 (Hölder cancellation)

The canceled group seems to be Z2 here.

For a planar point, the canceled group (primitive of Lin) is either

Circle × Z2, r.s = s.r or
Circle * Z2, with r.s = s.r-1

======
Par Laurent !
Doc, the mirroring you was talking about is also mirroring the time, not only the space ! They were right when they said that the time travel is not possible...

Anyway, here is the CAP:
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PostSubject: Re: Encyclopedia of Integer Sequences   Tue Nov 16, 2010 4:33 pm

Nick, did you tried :

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.



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