Transcript The Analytic Continuation of the Ackermann Function

The Analytic Continuation of the
Ackermann Function
What lies beyond exponentiation?
Extending the arithmetic operations
beyond addition, multiplication, and
exponentiation to the complex
numbers.
Overview
Very high level overview because of the amount of material
in multiple branches of mathematics.
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Complex Systems – A New Kind of Science
Arithmetic
Dynamics of the Complex Plane
Combinatorics
New Kind of Science
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Chaos beyond exponentiation.
Vertical catalog of complex systems.
Based on iterated functions.
Arithmetic and physics are two major roles
played by iterated functions.
• Iterated functions as a candidate for a
fundamental dynamical system in both
mathematics and physics.
Dynamics and Combinatorics
Arithmetic
• Arithmetic is part of the Foundations of
Mathematics.
• Ackermann function is a recursive function
that isn’t primitively recursive.
• Different definitions of the Ackermann
function.
• Transfinite mathematics
Systems of Notation for Arithmetic Operators
Operator
Spiral
Ackermann
Addition
a+b
ack(a,b,0)
Multiplication
a*b
ack(a,b,1)
Exponentiation
ab
Tetration
Pentation
Hexation
Knuth
Conway
ack(a,b,2)
a↑b
a→b→1
ba
ack(a,b,3)
a ↑↑ b
a→b→2
ba
ack(a,b,4)
a ↑↑↑ b
a→b→3
ack(a,b,5)
a↑↑↑↑b
a→b→4
...
Circulation
...
ack(a,b,∞)
a ↑∞ b
...
a→b→∞
Definition of Ackermann
Function
Let f(x) ≡ a → x → k and f(1) = a → 1 → k = a;
then
f2(1) = f(a) = a → a → k
= a → 2 → (k+1)
f3(1) = f(a → a → k) = a → (a → a → k) → k
= a → 3 → (k+1)
Negative Integers
a→ 1→2= a
a → 0 → 2 = 1 This differs from the historical Ackermann function where a → 0 → 2 = a
The next two are multivalued so the values on the principle branch are shown.
a → -1 → 2 = 0
a → -2 → 2 = -∞
Period three behavior
-1 → 0 → 3 = 1
-1 → 1 → 3 = -1
-1 → 2 → 3 = -1 → -1 → 2 = 0
-1 → 3 → 3 = -1 → (-1 → 2 → 3) → 2 = -1 → 0 → 2 = 1
The first indication that for negative integer the Ackermann function can be very stable.
for n>2
-1→k→n = - 1 → (k+3) → n.
-1 → 0 → n = 1
-1 → 1 → n = -1
-1 → 2 → n = 0
-1 → 3 → n = 1
Hypothesis
For 1 ≤ a < 2
a→∞→∞=a
Tetration Through Octation
Transfinite Ackermann Expansions
a → ∞ → k useful for creating a series of “interesting” transfinite number.
Transfinite nature of circulation:
2→2→∞
= 4
2→3→∞
= 2 → (2 → 2 → ∞) → ∞
= 2→4→∞
= 2 → (2 → (2 → 2 → ∞) → ∞) → ∞
= 2 → (2 → 4 → ∞) → ∞
= ∞
3→2→∞
= 3→3→∞
= 3 → (3 → 3 → ∞) → ∞
= 3 → (3 → (3→3→∞) → ∞) → ∞
= ∞
Tetration
• First objective is understanding tetration.
• What if tetration and beyond is vital for
mathematics or physics?
• With so many levels of self organization in
the world, tetration and beyond likely
exists.
Julia set for the map of
z
e
Tetration by period
Tetration by escape
Dynamical Systems
• Iterated function as a dynamical system.
• Analytic continuation can be reduced to a problem
in dynamics.
• Taylor series of iterated function. Most
mathematicians believe this is not possible, but my
research is consistent with other similar research
from the 1990’s.
• Iterated exponents for single valued and iterated
logarithms for multi-valued solutions.
Fa di Bruno formula
k1
n! f k  g1 
 gn 
Yn ( fg1 ,, fgn )   ( n )
   
k1!kn !  1! 
 n! 
kn
• Derivatives of composite functions.
• Fa di Bruno difference equation.
m
D f
n 1
m! D f  Df

  ( m ) 1
k1!km!  1!
k
• Hyperbolic case
• Maps are Flows
n
k1

D f
  

 m  1!
m 1
n



k m 1
  Dm f n
Classification of Fixed Points
• Topological Conjugancy and Functional
Equations – Multiple Cases for Solution
• Fixed Points in the Complex Plane
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Superattracting
Hyperbolic (repellors and attractors)
Irrationally Neutral
Rationally Neutral
Parabolic Rationally Neutral
Combinatorics
• OEIS – On Line Encyclopedia of Integer Sequences
• Umbral calculus and category theory.
• Bell polynomials as derivatives of composite functions.
Dm f(g(x))
• Schroeder summations.
• Hierarchies of height n and the combinatorics of tetration.
Exponential Generating
Functions
Hierarchies of 2 or Bell Numbers
Hierarchies of 3
Hierarchies of 4
xe
- Tetration as phylogenetic trees of width x
Schroeder Diagrams & Summations
Validations
• Deeply consistent with dynamics.
• f a (f b(z)) - f a+b(z) = 0 verified for a number of
solutions.
• Software validates for first eight derivatives and
first eight iterates. Mathematica software reviewed
by Ed Pegg Jr.
• A number of combinatorial structures from OEIS
computed correctly including fractional iterates.
NKS Summary
• Wolfram’s main criticism is inability of
continuous mathematics to deal with
iterated functions.
• CAs are mathematics not physics, many
non-physical solutions.
• “Physics CA” needs OKS for validation.
• CAs appear incompatible with Lorenz
transforms and Bell’s Theorem.
Summary
• Subject is in protomathematics stage, but becoming acceptable areas of
research; numerous postings on sci.math.research lately.
• Arithmetic → Dynamics → Combinatorics → Arithmetic
• If maps are flows, then the Ackermann function is transparently
extended.
• Suggests time could behave as if it is continuous regardless of whether
the underlying physics is discrete or continuous.
• Continuous iteration connects the “old” and the “new” kinds of
science. Partial differential iterated equations
• Tetration displays “sum of all paths” behavior, so logical starting place
to begin looking for tetration in physics is QFT and FPI. Tetration and
many other iterated smooth functions appear compatible with the
Lorenz transforms and Bell’s Theorem.