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- Puetz - Universal Wave Series (UWS)
- Description of the Universal Wave Series (UWS)
- Overall conjectures (guesses)
- System dynamics of the UWS
- Harmonics of [,D]UWS
- Frequency beats

- [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]).
- 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.
- 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.
- 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".
- Harmonics of UWS frequencies f(i) :
- ALL divisions of UWS frequencies by 3^n produce another UWS frequency
- ALL divisions of DUWS frequencies by UWS frequencies produce another DUWS frequency
- Divisions of UWS by DUWS frequencies do NOT produce another Union(UWS, DUWS) frequency.
- Divisions of UWS frequencies by [2,4,5,7,8,10,11,13,14,...] do NOT produce another Union(UWS, DUWS) frequency.
- Divisions of Union(UWS, DUWS) frequencies produce a ratio of either :

3/2 - all outside of Union(UWS, DUWS) frequencies

2 - all within UWS

3 - all within Union(UWS, DUWS)

- Divisions of Union(UWS, DUWS, HUWS) frequencies produce a ratio of either :

3/2 - all outside of Union(UWS, DUWS) frequencies

2 - all within UWS

3 - all within Union(UWS, DUWS)

- 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.- The "first frequency beat" produces either :

f(i-1).. (the lower UWS frequency), or

f(i-1)*2 (the "DUWS" of the lower frequency). - The "second frequency beat" produces either :

f(i-1)*5/3 < f(i-1)*2 or 33% less... than DUWS for f(i-1), or

f(i-1)*7/3 > f(i-1)*2 or 33% greater than DUWS for f(i-1)

Therefore, the second frequency beat DOES pose the issue of energy dissipation across non-DUWS frequencies which are a significant divergence from Puetz +- ~5% UWS matching criteria. - Below the "second frequency beat", frequency differences are on the order of 1/27, which I will assume is within the Puetz time series matching criteria of ~5% +-. (This should be checked rigorously with infinite series analysis).
- Differences between DUWS frequencies ???? NYET: always produce another frequency of Union(UWS, DUWS) ???
- Differences between Union(UWS, DUWS) frequencies ????
- Differences between Union(UWS, DUWS, HUWS) frequencies ????

- The "first frequency beat" produces either :
- Frequency splitting

"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 | ... |

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).

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.

- 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].
- Energy [dissipative, leakage] processes are "minimized by keeping [sympathetic resonance, beat, split, scavenge] effects within" the [H]UWS.
- 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!!!
- The current discussion does not yet examine frequency [inaccuracy, variability], which is an important omission.

- frequency [split, beat, scavange]
- sympathetic resonance
- weak signal inducement of [event, phase, state] change of chaotic systems and imprint of strong periodicity
- probably many examples in electrical engineering signal processing textbooks
- also chaotic NNs

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]

[*, /] 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)

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, ????

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

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.