Function composition of symmetries -n, 1/n for FLT
Subj: Re: Non-associative and non-distributive operations
Date: 19 Oct 2002 (sci.math)
From: Nico Benschop
Orgn: Digital Research : Finite & Associative
----------------
nhp wrote:
>
> Could somebody give an example of some "reasonable" operation
>
> a) not obeying the associative law (i.e. (x*y)*z != x*(y*z))
> b) not obeying the distributive law (i.e. x*(y+z) != x*y + x*z)
> c) not obeying neither associative nor distributive law
>
> Of course one can always introduce some quite arbitrary * and + such
> that the laws are not obeyed, but I'm looking for some operations
> which are actually used in maths or at least are somehow "reasonable".
>
> As simple as the laws are, they are (or maybe just because of that) at
> the heart of the maths. I think knowing some examples about what the
> laws exlude would give further insight into what these laws and
> calculus is all about.
The distributive law holds for one operation (*) over another (+)
if (*) is defined as repetition of (+). Think of the normal
arithmetic operations (^) over (*), and (*) over (+) for integers.
To get a feeling for how restrictive 'associative' property is,
notice that it is necessary for the important concept of 'iteration':
e.g.: (aa)a = a(aa) is denoted a^3 only in this associative case.
Imagine an order n closure of set S under some operation (.) defined
by its n x n composition table. There are n^{n^2} ways to fill such
table, of which n! row/col permutations yield isomorphism's.
Now try to find the % of such square tables (closures) that are in fact
associative. Then you'll find for not too large n already (say n=10)
an almost negligible fraction is indeed associative. Meaning that
the 'associative' assumption for an operation: semigroup S(.) is
a VERY strong restriction (something like 'linear' for networks),
implying a lot of 'structure'... For non-associative "de beer is los"
(anything goes;-) - one example is exponentiation (^) over (+),
such as FLT (additive statement with exponential terms):
-- NB - http://piazza.iae.nl/users/benschop/sgrp-flt.htm
(on semigroups & function composition in arithmetic)
Function composition of symmetries -n, 1/n for FLT
(news:sci.math . . 4apr98)See On discovering the triplets (mod p^k) . . . (double-click)
This triplet structure is used in a direct proof of FLT (Fermat's Last Theorem), The smallest example is Z (mod 3^2) = 2* = {2 4 8 7 5 1} = {2 4 -1 -2 -4 1} mod 9. This 3-loop holds for each n<>-1 in G mod p^k, for each prime p, and all k>0, except for the cubic roots of 1 mod p^k (p=1 mod 6) where the looplength is one: a+1 = -1/a (a<>1), and in a special case where 4 divides p-1 (so p=1 mod 4). . . . Moreover, the Hensel lift (of extending a mod p^2 solution to mod p^k for any k>2) can be "broken" by this a+1= -1/a solution, because the Exponent p Distributes over a Sum: (a+1)^p = a^p + 1 = a + 1 (mod p^k, EDS property in Core . : . see my paper on " Triplets ..." , ref[1] on my homepage). The clue, to prove no loop longer than 3 exists in ring Z(+, .) are its two symmetries (= automorphism of order 2), seen as functions, namely: and : Inverse function I(n) = 1/n for multiply ( neutral element '1' ). "Simple comme bonjour", would Fermat have said... ... although for p-th powers the first triplet is at p=59, and arithmetic doodling mod 59^2 in those days with his PC (Pascal Calculator = "Pascaline") is a bit unlikely. -- Yet, we don't need p-th powers to see the triplet structure of arithmetic, as shown above! Moreover, for p=7 the cubic roots occur already, which he very well could have found (for inequality FLT case1 they should be the only solutions mod p^2, which they are NOT). Check out http://www.iaehv.nl/users/benschop/ferm.htm And the same 3-loop holds for ALL four possible compositions of these three elementary arithmetic functions C(n), I(n) and the successor S(n)=n+1. Notice that I and C commute, so 4 rather than 3! = 6 such "dfs" functions arise (dual folded successor functions). but requires more powerful function composition (semigroups). As it were .: You need a diamond (sgrp) to cut steel (arithm). For instance the above function is SIC(n)= -1/(n+1) = SCI(n), where function composition is normally from left to right -- following this nice notation advocated by Clifford/Preston in the reference work: "The Algebraic Theory of Semigroups" (AMS Survey #7, 1961) You see that . : . looplength three = the number of symmetries + 1 = the number of operations + 1.
( . . I think this is not a coincidence, but a necessity . . ) The extension to inequality for integers follows from a variant of the Exponent p Distributes over a Sum (EDS) property of this solution of FLT case1 in residues ("two terms of a triplet are in Core" -- see my homepage ref[1] on the general Triplet structure of Arithmetic mod p^k, not only for p-th power residues). Constructive comment is welcome.
| |
--------( Do not associate this title with Science = Sick ;-)--------- --------( Rather: an exercise in function composition )--------- Consider in arithmetic ring Z(+, .) the two basic symmetries -n and 1/n as functions, in fact automorphisms of order 2, of Z(+) resp. Z(.): And denote the successor function as: S(n) = n+1. Notice that IC = CI (commute), but S does not commute with I or C. (so indeed SCI = SIC ;-) Then compose all three functions in all possible (four) ways: n(SCI) = -1/(1+n) n(CSI) = 1/(1-n) n(ISC) = -(1+ 1/n) Watch notation: function composition is from left to right. Call such composition a "dfs" (dual folded successor) function. . . . . The third iteration of each dfs function is the identity function E . . . . . . ( nE = n, for all n<>0,-1,1 ).
Proof: . . Let F_i be the i-th iteration of a function F. Then show: (dfs)_3 = E.
. . . n(SCI)_2 = -1/[1 -1/(1+n)] = -(1+n)/[1+n -1] = -(1+n)/n = -1/n -1 . . . n(SCI)_3 = (1+n) -1 = n . . . QED. Similarly, verify the other three dfs functions to have period 3.
An interesting consequence of this "3-loop" property of arithmetic is
that for residues mod p^k (prime p>2, k>1):
. . . . a+1 = -1/b --> b = -1/(a+1) (provided division by zero is avoided, so e.g: a+1 <>0, etc.)
Basic triplet example is in G(.) mod 9 = 2* = {2, 4, 8, 7, 5, 1} . .
where 8= -1, 7= -2, 5= -4 (mod 9).
. . . . . 1(SCI)_*: -1/(1+1)=4, -1/(4+1)=7, -1/(7+1)=1, with abc=1.4.7=1 mod 9 Note : maximal period=3 is the number of symmetries +1 (coincidence?-)
. . . For this function composition result applied to a proof of FLTcase1,
Re: proof of Goldbach's Conjecture . . . sci.math-9sep99
|
Subject: Re: proof of Goldbach's Conjecture
Author: Nico Benschop
Date: 9 Sep 99 07:04:03 -0400 (EDT)
seminole1215@my-deja.com wrote:
>
> Why you guys are wasting the time on this problem, I would never
> believe anyone in this planet could prove the Goldbach's Conjecture
> with 6-page, not even with 60-page. Much harder, guys. Handwaving
> stuff won't help and old tools with which people proved the cases of
> (1+n) (n>1) couldn't be used for case (1+1).
> I truely believe we need brand-new tool to task it. ...[*]
> It might take a couple of decades and even centuries. ...[&]
>
Re[*]: Not quite: rather USE tools that are wellknown,
such as the algebra of function composition
[ associative, but not commutative: f(g(x)) =/= g(f(x) ]
to solve hard problems in arithmetic.
(like on powersums: Fermat, Waring; or primesums: Goldbach;-)
For instance: the two symmetries of arithmetic: complement -n
under (+) about "0", and inverse 1/n under (.) about "1",
have a fascinating 3-loop property that only can be seen under
function composition. Namely, call them 'C' and 'I' respectively,
and let 'S' be the Peano successor function n --> n+1.
Now consider function SCI (from left to right apply S first, then C
and lastly I), then you'll easily verify that this function 3 times
applied in iteration, yields the identity function E: n --> n.
(do not divide by 0, so some restrictions hold: n<>0, and n<>-1 ):
So: n(SCI)= -1/(n+1) applied twice more to itself:
n(SCI SCI)= -1/{-1/(n+1) +1} = -(n+1)/{-1 + n+1} = -(n+1)/n
n(SCI SCI SCI)= -1/{-(n+1)/n +1} = -n/{-n-1 +n} = n.
Funny, magic, how come: a basic 3-loop linking the fundamental two
symmetries of arithmetic and the Peano successor function ...!?
E.g: this is the clue to all solutions of x^p + y^p = z^p mod p^k,
namely the "Triplet": a+1=-1/b, b+1=-1/c, c+1=-1/a, with abc=1 mod p^k
basic 3-loop structure of residue arithmetic mod p^k (prime p>2),
(for ALL residues coprime to p) & a 'sideline' to integers: FLTcase1,
breaking the Hensel-lift by taking into account the 'carry':
the p-th power of a k_digit number (base p) has upto kp digits -
------- the carry makes the difference, for FLT ;-)
Indeed: an OLD and well known tool (function composition) applied
in a NEW way! -- That's why I find: mathematicians are sitting on a
goldmine (semigroups = associative algebra, especially in_the_finite)
without really knowing it... (like Shannon in 1938 suggested to apply
Boolean Algebra - a commutative & idempotent form of arithmetic -
to the specification and design of combinational logic circuits:
Boole's work was some 90 years old 1848: his monograph precursor
of "The Laws of Thought" 1854). For sequential logic synthesis (FSM:
Finite State Machines = computers) no such fundamental & practical
tool has as yet been developped, although its basis: semigroup algebra
(=function composition) is already existant since 1928 (Shushkewitch).
Re[&]: It need not necessarily take that long: IF we, open_minded
engineers, scientists, and yes: some mathematicians - (although
the latter have the disadvantage of the specialist/expert:
single focus;-) - USE the tools already developed by previous
generations, ...the older & simpler the better...
Number theory without fundamental use of function composition algebra
[semigroups, like Z(.) analysed additively, including non-cmt, finite,
with divisors of zero, ..&c] I would dare to call not quite complete,
missing an essential concept.
========= It takes Steel to cut Wood,
It takes Diamond to cut Steel... ========================
where: Wood = All Practical Purposes (APP in Science & Engineering)
Steel = Arithmetic, Calculus, Set_theory (classical methods)
Diamond = Associative Function Compostion (Semigroups)
--
Ciao, Nico Benschop - http://www.iae.nl/users/benschop
http://www.iae.nl/users/benschop/campaign.htm
http://www.iae.nl/users/benschop/func.htm
http://www.iae.nl/users/benschop/fewago.htm
http://www.iae.nl/users/benschop/scimat98.htm
Subject: Re: logic, combinations, permutations
Date: Thu, 3 Aug 2000 12:48:21 GMT
From: Nico Benschop
Org: Research
Newsgrp: sci.math
---------
Niek Sprakel wrote:
>
> Is there any coherent, consistent and complete theory which relates
> permutations, combinations and logic?
> I reckon permutations are embeded among multinomialcoefficients and
> propositional logic is closely related to binomialcoefficients.
>
A good context for these algebra's (of permutations, transformations,
sets, arithmetic, combinational- and sequential logic, FSM: state-
machines) is given by the common property of the corresponding
operations: associative.
Hence: semigroups (= associative algebra of functions)
is their common context.
--
Ciao, Nico Benschop -- http://piazza.iae.nl/users/benschop/sgrp-flt.htm
simple sgrps as FSM: http://piazza.iae.nl/users/benschop/c-ranksm.dvi
integer state machines http://piazza.iae.nl/users/benschop/ism.htm
|
| -- N.F.Benschop -- July 1998 -- |