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Essential Motivation (C.Shannon)
John Glenn (77), Shuttle #92: Discovery - 7nov1998 @
Charles Lindbergh (25), Spirit of St.Louis - 21may1927 @
Le Bourget,
. . Home email: . . AmSpade Research
Edouard Manet
(1882): Chrystal vase + clematis
Some WWWisdom on Logic, FLT, Goldbach,
Automata "State Machine decomposition
by Right-congruences" . . . . (Asymmetric time) Cantor's uncountable
strings A* over alphabet of |A| > 1 letters (sci.math
feb'02) A simple FLT proof _is_ possible : via the cubic roots of 1 mod p^k. . . . . . . Goldbach elementary proof . . . . . . FLT elementary proof (p169-184) |
- My new
BOOK - preface.htm ,
Cover
Order
(19.50 EU) at publisher
abc.nl
( See ACM Computing review, and Zentralblatt.Math
review , If stuck at Closure: use the Carry . . . Residue-and-Carry method to prove FLT and Goldbach
Symmetric and Planar Logic Synthesis . . . Each boolean
function BF_n for n<5 is planar |
Homepage2 on Astro-physics
(redshift, foton-decay,
ether): Truth depends on Axioms ::: Axiom = Self-evident = a matter of Opinion
= Belief. On New Ideas in Science (and resistence to them), by Thomas Gold (1989). Paul Marmet: "...explanations which are
compatible with Newtonian and semi-classical physics. BigBang is wrong, by John Kierein
Hubble redshift as
non-Doppler effect: - Some of my papers on Arxiv (at front.math.ucdavis.edu) "Commonsense-in-Science Manifesto" 1. . . A wave phenomenon requires a medium, 2 . . . . EM waves (light) propagate in space, 3. . . Perpetuüm mobile (lossfree)
processes do not exist, 4. . . Speed of wave propagation depends on medium density, 5. . . With matter (electron) as closed wave (closed photon) [*]: 6. [!] . . If ether density around heavy body decreases as 1/r 7. The constancy of lightspeed c holds for
all frequencies with accuracy 10^{-21}. . . Nothing compels it to be the absolute maximum speed (as assumed by Einstein). . . Similar to the speed of sound in air: a projectile with enough energy can break . . though the sound barrier (e.g. a jet airplane). So a neutrino with enough energy can . . break through the lightspeed barrier, possibly with a V-cone shockwave. . . There is no connection with causality problems, time reversal, twin paradox, etc. [*] Is the electron a photon with toroidal topology? [*] Quasars, Big-Bang &
Cosmological constant by dr. Paul Marmet |
"Big Bang, Black Holes, and
Common Sense"
by David Pratt
Redshift History by R. Gray and J. Dunning-Davies (
"A new non-Doppler Redshift" by dr. Paul Marmet.
On Fermat's marginal note, a
suggestion N.F.Benschop (NMC-33, Apr.1998)
My father's diary on his Birma
Railroad experience 1942-'45 (in Dutch).
Buy
hardcover copy (Lulu.com) 75 pgs
or Softcover.
Curiosity and Search . . . . . "An open mind is
a joy forever "
. Inertia . . . . . . . . . . . . . . . . "
Climbing
. . Generalize . . . . . . . . "If a problem is too
hard, generalize it "
. . . Simplify . . . . . " We have to fight uphill to
rediscover the obvious "
. . . . Coincidence . . . . . . . . . " Life is fighting
entropy "
My CV . . . Work . . . Play . . . ff . . finite fun .
. (reflections in the finite)*
. . . . . . Intuition : . . Even if you don't see it, you still can smell it !
. . . . . . Bamboe . . . Chinese brush painting
Remembering: - (NB:
28-july-2002)
Claude Shannon
Fermat, Waring,
Goldbach -:- view Arithmetic as Function composition (semigroups) :
. . . Additive Number Theory : hard problems, solved via semigroups
Z(.) mod m_k , and . . .
"Additive structure of the
Group of Units mod p^k ,
with Core and Carry concepts for extension to Integers"
- - - Triplet structure, and direct FLT proof (16 pgs), published
Nov.2005 in:
AMUC - Acta
Mathematica Univ. Bratislava
Péano- and Cantor- sets and their generation: sequential dimension
The sawtooth of Evolution :
. . Extremes increase _ - + * ^ and then collapse \___ . . &c (with
positive Carry ?-)
Child's arithmetic ( = exponential ;-( . . . Stable
economic growth of 3 % per year,
. . . then *(1.03)^n yields economy *10 ... *100 ... *1000 after n = 78
... 156 ... 234 years !
NB: . . . One Generation =
26 yrs ==> Your grandchildren live 10 x dense than you . . .
Humor / Relegion
"If God created us in his own image, we have more than reciprocated."
— Voltaire (1694-1778).
Links (en Rechts;-) + Discussions . . . my 8 papers at : de.arXiv.org
Among others :
- Finite Semigroups of Constant Rank, and the five
- On Log-arithmetic, Fermat's Small Theorem, and the powers of 3 (mod 2^k).
- Extending Fermat's Small Thm to r^{p-1} mod p^3,
for divisors r | p +/- 1.
My cup-a-soup :
^ * + - . : . - + * ^
Once upon a time there were three kinds of math-objects : Sets, Numbers, Functions
(between Sets)
Set |S|=n has 2^{n} subsets, n! permutations, n^{n} transforms (say n=5 : 2.2.2.2.2
< 1.2.3.4.5 < 5.5.5.5.5)
Sets (logic) compose by intersection( /\ ) and union( v ) which are both
idempotent : a/\a=a, a\/a=a.
Numbers you can add(+) and multiply(.) which both are commutative:
a+b=b+a, a.b=b.a
Functions (seq_logic) compose by sequencing (
) which is associative: a(bc)=(ab)c.
Including logic and arithmetic, which are also associative, we get Semigroups,
making a triple sandwich of logic < arithmetic < function-composition,
and the 3 algebra's are idempotent < commutative < associative.
Like order and equivalence: reflexive | symmetric | transitive.
The two sides of the same coin : objects form semantics,
sequential behaviour of composition rules is the syntax.
- Exponentiation (^) falls out of the boat for misbehaving -
It is not idt, cmt,
assoc; ---- It distributes over (.) but not over (+)
So it becomes really interesting when you combine (+) and (^) as in x^p + y^p = z^p.
with no integer solution for p>2: the powersum
yields no p-th power, but what then ? [6]
++ It turns out that p-th power residues mod p^k are the best additive residue base ++
Symmetries -n and 1/n mod p^k,
with n+1= -1/n yield FST --> FLT [3], and
other stu ff.
M S D :
More specific designs: . Powersums [3 1
5 6 7] . . 3* code [21] . . FSM seq_logic
synthesis [2]:
[-4] Internal State of Computer Science (ICS73.pdf)
. . . . . . . My original motivation for Digital Network Theory <=>
Finite Semigroups
[-3] Re: unify all branches in math - and some math history (sci.math 19feb02)
[-2] Re: Problem with Cantor's diagonal
argument
. . . Cantor's Diagonals generate powerset 2^N by
permutations of w x w table rows (sci.math 15feb02)
[-1] Fermat's Anti-Closure n^p (+) as
Generator (.htm) . . . . . . . . . . . . news:sci.math
- 28nov98
[0] Function composition of symmetries
-n, 1/n for FLT (.htm) . . . . . . . . . . . . sci.math - apr98
[1] Triplets and Symmetries of
Arithmetic mod p^k (.pdf) . . . Semigroups,
Arithmetic (Fermat Last Thm FLT)
[2] On the structure of Constant-Rank State
Machines (.pdf) . . Sgrps and Sequential Logic Synthesis
[3] Fermat and the Cubic roots of 1 mod p^k (.html) . .
. . . . . .Breaking the Hensel lift . (1 page clue)
[4] On the triplet structure of
arithmetic mod p^k,....
[5] On Fermats'
marginal note: a suggestion (.pdf) . . . Cubic roots: a^3=1 mod p^k (marg-abs.htm)
. . . . Quotes E.T.Bell
(.htm) : . . . . . . . . . . . . On abstraction, specialisation and divergence.
[6] On powersums
representing residues mod p^k(.pdf) . . . . " Waring-for-residues
" (nw-abstr.htm)
. . . . . -------- Each residue mod p^k is the sum of
at most four p-th power residues --------
[7] Arithmetic with squarefree
modulus, its lattice of groups,
. . . a primesieve, and Goldbach's
conjecture(.pdf) . . . Goldbach
proof (old version 2002):
[8]-- Fermat, Waring,
Goldbach -- (.htm) . . . . . . . . . . . . . . . . . .
via Semigroups Z(.) mod m_k
[9] One more crank (.htm) sci.math
- may98 . . . . . . . Hasse-Minkowski thm and Selmer's group
[10] How diagonal is Cantor's diagonal ? (.htm) . . . A Bootstrap process generating
the reals.
. . . . . . . . . . . sci.math 5jun98 . . 6jun98
[11] Semigroups and FLT (.htm) . . Z(.,+) mod m . . . . . .
Function composition linking (+) and (^)
[12] New Math (base p) applied to FLT (.htm) .
. . . . . . . (Fermat?) computing a^3=1 mod 7^2.
[13] Re: Orthogonality (.htm) . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . sci.math - 3nov98
[14] Re: Primitive roots of Unity (.htm) . . . . . . . . . . . . . . . . . .
. . . . . . sci.math - 3nov98
[15] -- Stumbling over Aladin's lamp .. sci.math 13nov98 .
. . . . . . . . . sci.math 10nov98
[15'] A simple direct FLT proof after all
? (despite the Hensel lift) . . . . sci.math
- 1aug00
[16] Fermat's anti-closure n^p(+) as generator / BOOA constructor . . . sci.math
- 4dec98
[17] Re: Finite versus Infinite (a matter of taste) . . . . . . . .
. . . . . . . . . . sci.math - 16feb99
[18] Re: Why the attack on Cantor's
Theory? . .
from Cardinals to Types . . . sci.math 27feb99
[19] Easter primes do not exist: 2^p != 2
mod p^3 (prime p)
sci.math 12apr98, and :
[19b] Divisors r of p-1 (resp. p+1) have
distinct r^{p-1} mod p^3 . . Carry-extend Fermat's Small Thm. (.pdf)
[20] The role of prejudice in the exact
sciences . . . a
non-discussion on nmbrthry.list (14mar99)
[21] My +/- 3*2* Binary Multiplier . . . . US patent 59.23.888 (13 july 1999):
. . . 3 is semi-primitive root of 1 mod 2^k, Log-arithmetic (dual base 2,3) .. (.pdf)
[22] Sequential dimension of finite and
infinite closures: N vs 2^N . . . sci.math
7oct99
[23] ISM : Integer State Machines . . . . . . M(Z,A) state set Z,
inputs A={+1, -1,swap(2)}
[24] Discovering the Triplets in Z(.) mod
p^k . . (a+1)b=(b+1)c=(c+1)a = -1, abc = 1 (mod
p^k)
[25] On FSM-decomposition and Semigroups (= sequential closure): part I . . . . . part II
[26] On Reals
(as un-real concept)
: . . One can specify only countably many reals anyway!
__ L S D : Less simplistic
directions [*.html] : . . . a wider view - beyond Logic and Arithmetic :
[a] . Cubic roots of 1 . . . . . . . . . . . . . . . . . Triplets
/3/ and Symmetry /2/ of FLT mod p^k
roots
[b] . Residue and Carry . . . . . . . . . . . . . . .
. . for Closure and Generation ( confirm and explore)
[c] . A day with Fermat in Toulouse . . . . . . Did he find the cubic
root solution of FLT mod 7^2 ?
[d] . I-Ching (symbol 59 and text) . . . . . . The Solution .
(of a stuck-at situation: whip-up water)
[e] . Bibliotheca Philosophica
Hermetica (Amsterdam)
. . . "
[f] . I-Ching : 8 basic icons , Yi-Jing info
. . Chinese philosophy: Balanced Change, Cyclic Time
[g] . On Dynamic Balance and Bootstrap . . . On a balanced psychology
(Maslow and Spangler)
[h] . Weighing Cantor's Evidence . . . . . . . . . . . . . . . On
math, EE-practice and real infinity
[i] . Red-shift of fotons
by decay: Ether after all? . . . . . . . . . . On lossy light
propagation.
[j] . Strange dominance of math in world
model . . . . .
. . . . Maslow's "hammer and nails"
[k] . Re: Pi is no piece of Pie . . . . . . . . . . . . . Finite
atomism vs Infinite partition (sci.math)
[l] . On controversies . . . . . . . . Finite/Infinite,
Discrete/Continuous, Tradition_/Modern_units
[m] . Something Rotten . . . . at the Core of Science? (Trends
in Pharmacological Sciences)
[n] . On Digital physics (Fredkin) . . . and Cellular automata (Steven Wolfram)
[o] Bush' Mission to Mars (TIME 23feb04): " Sending humans to Mars
sounds
. . . fantastic, like something to strive for. But the problem is that the
mission will
. . . just give us another world to mess up. Untill
we have a complete change of
. . . attitude and behaviour, let's stay away from other planets!" (L.Fourie, S-Africa)
---- "Why should we spend vast amounts of money to reach Mars when the
chance is very
---- high that the mission does not provide anything to improve our lot here on
Earth?
---- Why not allocate those billions for "home improvement"? How
about curing cancer,
---- aids and other diseases; cleaning our water, air and soil; feeding hungry
nations.
---- [NB: Or making the
---- Why confront the hazards of outer space when we should be fighting
homelessness,
---- poverty and unemployment on Earth? (C.Brunish,
---------------- if stuck@closure (mod
...) : use the Carry ! ----------------
--- from Numbers {n+p, n.p, n^p} to Sytems Z(+)(.) and (^) mod m ---
-------------- FASYL . : . Finite
Associative Systems
Logic ---------------
All updates |
NB : At a
philosophical & common-sense level, Godel's
result came surpisingly late,
. . . just saying that every theory (as transitive closure of a finite set of
axioms,
. . . using logic/causality as generating principle) has its context.
So : TRUTH is relative to your assumptions (axioms), and math in
particular
. . . . . . . does not yield anything really new beyond their transitive
closure !
[V] . Campaign : for the Cubic roots of Unity, breaking Hensel's lift (re: FLT), and
to promote finite transformation Semigroups
and functions for Logic, Arithmetic, FSM's
[2].
--- If you like this direct approach [3] to FLT (case_1): tell a
friend (or even me ;-)
--- And if you do not like it : please tell me why;
. . . certainly if you can disprove the EDS argument [3] for the cubic
root solution.
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