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Howell - preliminary frequency analysis, Fourier series as origins of Puetz UWS

These analysis are very crude, and probably wrong. But it's a start...


Table of Contents

Summary of overall conjectures (guesses)

  1. [UWS, Fibonacci] series are among the simplest fractal systems (perhaps in a loose sense?). Their first levels down are (Puetz UWS [0.333, 0.667] levels ~= Fibonacci [0.236, 0.618]).
  2. The "core factor of 3" UWS series must be a reflection of an important [direct, indirect] causal driver(s) across enormous real [time, length] scales. That driver(s) has yet to be established.
  3. The UWS series frequencies cannot be absolutely constant if the UWS is to represent natural time series, even though the long-term "average" frequencies are assumed to be fixed. For example, the sunspot half-cycle is 11.1 years on average since ~1710, and ranges from 8.7-12.2 years (these are not accurate figures - from memories >10 years old???). Only 2 or 3 times has the half cycle been close to 11.1.
  4. The UWS is relatively resistant to energy [loss, dissipation] across non-UWS frequencies, by processes such as frequency [split, beat, scavenge]. The latter processes do spread energy across UWS frequencies, whereas dissipation across non-UWS frequencies is relatively "seconday, minor".
  5. Harmonics of UWS frequencies f(i) :
  6. Frequency beating (differences) of adjacent UWS frequencies (f(i), f(i-1)) :
    Note that beats are of progressively longerperiod as the differences between frequencies is smaller.
  7. Frequency splitting

Description of the Universal Wave Series (UWS), acceptance criteria

Puetz "Universal Wave Series" ( UWS) - 3 [times, divide]=[*, ] current frequency
"Double Universal Wave Series" (DUWS) - 2*UWS series
"Half Universal Wave Series" (HUWS) - 1/2*UWS series

Here are tables of sequential series of [,H,D]UWS, Union(UWS, DUWS) frequencies.
UWS ... 1/27 1/9 1/3 1 3 9 27 ...
DUWS ... 2/27 2/9 2/3 2 6 18 54 ...
NOTE : DUWS is also a "factor of 3" periodicity series, like UWS.
DUWS = 2*UWS

Let U1 = union(UWS, DUWS), U2=Union(UWS,DUWS,HUWS)
U1...1/27 2/27 1/9 2/9 1/3 2/3 1 2 3 6 9 18 27 54...
U1(i+1) / U1(i) ...2 3/2 2 3/2 2 3/2 2 3/2 2 3/2 2 3/2 2 3/2...
U2...1/27 1/18 2/27 1/9 1/6 2/9 1/3 1/2 2/3 1 3/2 2 3 4.5 6 9 13.5 18 27 54...
U2(i+1) / U2(i) ...2 3/2 2 3/2 2 3/2 2 3/2 2 3/2 2 3/2 2 3/2...

Puetz "acceptance criteria" that a time series that matches [H,D]UWS is within ~5% +-, if I remember correctly. But was this on a log scale or linear? (I assume log scale here). Even for 3/2 factor for U1(i+1)/U1(i), 5% is a very tight (significant) criteria. It is somewhat reminiscent of the standard 95% confidence interval in statistics. Puetz explains this, and his special statistical tests as well as standard tests, in his book.

Note that Puetz, and others, refer the [H]UWS sequence as a fractal system, which is strange to me. After a quick look at the definition of fractal, I cannot confirm that, but it seems reasonable (to check again at some time in the future).


System dynamics of the UWS

The Puetz "acceptance criteria" as member of UWS for a periodicity is that the periodicity must be within ~5% (if I remember correctly) of one of the official Puetz "Universal Wave Series" (UWS) or "Half Universal Wave Series" (HUWS) lambda (frequency or period). I ignore the "Double UWS" for much of this webPage, as it appears that Puetz is noww more focused on the HUWS instead. But was this on a linear or log scale? (should be log scale - need to check). Even for a 3/2 mnimum factor (HUWS), 5% is a very tight criteria, somewhat related to the standard 95% confidence interval concept that is commonly used.

This acceptance criteria tells us something important : if Puetz [H]UWS frequencies make up a non-ramdom portion of all known periodicities, then it really begs the question of why. Possible phenomenological explanations are discussed in , but on this webPage alysis is uniquely focussed on the dynamical implications of the [H]UWS, irrespective of phenomena that my drive it.


Key assumptions :

  1. UWS must reflect any underlying [direct, indirect] causation, as well as the expected dynamic freeuncy effects [harmonics, resonance, beat, split, scavenge, imprint of chaotic system].
  2. Energy [dissipative, leakage] processes are "minimized by keeping [sympathetic resonance, beat, split, scavenge] effects within" the [H]UWS.
  3. The previous point applies in particular if most other frequences "leak" into the UWS because they do NOT have that property. Hurst [*/2] sequences are a separate example that won't crrespond to UWS cycles. Note that the HUWS is NOT a Hurst sequence!!!
  4. The current discussion does not yet examine frequency [inaccuracy, variability], which is an important omission.

System dynamics of the UWS


how to explain 10^25 factor (25 orders of magnitude) scalaing of UWS periodicity?
perhaps explain [,non-]adherent process contexts
eg prime # years ciracada population explosions

non-resonant period inducements?
car [vibration, noise]
chaotic system & memory capacity
frequency beating [Timo Nimura, Paul Vughan]


Harmonics of [H]UWS

Harmonics of the UWS : consider integer [*, /] cart [2, 3, 4, 5, 6, 7, ...]
[*, /] 3 applies to all [H]UWS (already in U1)
*2 is there ONLY for UWS (yields a DUWS member), but not for DUWS
/2 is there only for DUWS (yields a UWS member), but not for UWS

To get "full benefit" from frequency effects, I will now look at the expanded series :
U2 = union(UWS, DUWS, HUWS)


Frequency beating

Frequency-beating is the result of difference between two periodicities (linear scale, not log).

f_beat = (f1 - f2)


UWS frequency beating

f_beat(1) = f(i) - f(i-1) = f(i)*2/3 = f(i-1)*2 (i.e. DUWS)
f_beat(1,1) = f_beat(1) - f(i-1) = f(i) - f(i-1)*2 = f(i)*(1 - 2/3)
= f(i-1) (UWS)
f_beat(1,2) = f_beat(1,1) - f(i-1) = f(i-1) - f(i-1)
= 0 (UWS) -> no beating, only f(i-1)
f_beat(1,3) = f_beat(1,2) - f(i-1) = 0 - f(i-1) -> no beating, only f(i-1)
= 0 (UWS)
Therefore, UWS alone does not produce dissipation via self-beating.


One might possibly consider HUWS and DUWS as the simplest self-beating frequency sequences of UWS. For HUWS :

f_beat(2) = f(i) - f(i-2) = f(i)*(1 - 1/9) = f(i)*8/9
f_beat(2,1) = f_beat(2) - f(i-1) = f(i)*8/9 - f(i-1) = f(i)*(8/9 - 1/3) = f(i)*5/9
= f(i-1)*5/3 (< f(i-1)*2 or 33% less than DUWS for f(i-1))
f_beat(2,2) = f_beat(2,1) - f(i-2) = f(i)*8/9 - f(i-2) = f(i)*(8/9 - 1/9)
= f(i)*7/9
= f(i-1)*7/3
so (f(i-1)*2 < f_beat(2,2)), or 33% greater than DUWS for f(i-1))
The f_beat(2,j) series does pose the issue of self-beating frequencies of significant divergence from Puetz +- ~5% UWS matching criteria.


For DUWS :

f_beat(2) = f(i) - f(i-2) = f(i)*(1 - 1/9) = f(i)*8/9
f_beat(2,1) = f_beat(2) - f(i-1) = f(i)*8/9 - f(i-1) = f(i)*(8/9 - 1/3) = f(i)*5/9
= f(i-1)*5/3 (< f(i-1)*2 or 33% less than DUWS for f(i-1))
f_beat(2,2) = f_beat(2,1) - f(i-2) = f(i)*8/9 - f(i-2) = f(i)*(8/9 - 1/9)
= f(i)*7/9
= f(i-1)*7/3
so (f(i-1)*2 < f_beat(2,2)), or 33% greater than DUWS for f(i-1))
The f_beat(2,j) series does pose the issue of self-beating frequencies of significant divergence from Puetz +- ~5% UWS matching criteria.

Below f_beat(2), frequency differences are on the order of 1/27, which is within the Puetz time series matching criteria of ~5% +-



Union(UWS, DUWS) frequency beating - Because the UWS is immune to self-beating, the combination of UWS & HUWS, ????

Frequency squashing by dissipative processes

Is this different from frequency scavenging?

Frequency scavenging

Frequency-scavenging is the result of ??.

As a first guess, it is the interaction between the following that is of interest :
(1 - x/OR[3,2]) * OR[3,2] * [H]UWS

Ied to plot this!



... note finished typing out notes from 23Apr2022


Non=[H]UWS prime number cycles

Prime numbers that are NOT part of the [H]UWS series or its harmonics include [5,7,11,13,17,19,23,27], Note that the 5% criterion is a period of 20, so those larger than that may not be as important? (guess) Should this also include products of non-[H]UWS primes and [H]UWS periods : [14,15,21,..]?

Primes are mentioned here given their overall importance in math. Whether that corresponds to a key issue for Puetz [H]UWS or Fourier series is another matter.