In formal mathematical terms of geometry/topology/homotopy/homology, let’s try proving that a wavenumber=0 cycle of east/west direction inside an equatorial toroidal-shaped waveguide, can only be forced by the Z-component of a (x,y,z) vector where x,y lies in the equatorial plane.
To address this question, let’s dissect the components involved and prove within the constraints of geometry, topology, homotopy, and homology, focusing on valid mathematical principles.
Similar to the LTE modulation in Chapter 12, modulation of a periodic impulse can lead to complex waveforms.
Find behaviors that are periodic impulse trains that are modulated by another periodic process, leading to another waveform with strong subbands.
ChatGPT: The description you provided suggests a phenomenon where a periodic impulse train, a series of regular, sharp signals, is modulated by another periodic process, leading to a composite waveform characterized by pronounced subbands. This can be observed in various fields, including signal processing, telecommunications, and even in natural systems. Here are a few examples where such behaviors might be observed:
Frequency Modulation (FM) in Telecommunications: In FM, the frequency of a carrier wave is varied in accordance with the amplitude of a modulating signal. If the modulating signal is a periodic impulse train, the resulting FM signal will exhibit a spectrum with sidebands at multiples of the modulating frequency. These sidebands can be viewed as subbands in the spectrum, resulting from the modulation process.
Amplitude Modulation (AM) in Radio Broadcasting: In AM, the amplitude of a carrier signal is varied in proportion to that of a modulating signal, which could be a periodic impulse train. This modulation results in a waveform where the carrier frequency is accompanied by sidebands (subbands) at frequencies equal to the sum and difference of the carrier frequency and the modulating frequencies.
Biological Rhythms: In biological systems, certain rhythms or signals can modulate others. For example, the circadian rhythm (a roughly 24-hour cycle) can modulate other physiological processes, leading to patterns that have periodic qualities modulated by the circadian rhythm, resulting in complex waveform behaviors with distinguishable sub-patterns or subbands.
Vibration Analysis in Mechanical Systems: In mechanical systems, a periodic impulse (such as that from a rotating imbalance) can modulate the vibration signatures of other components of the system. This can lead to complex vibration signatures that contain subbands corresponding to the modulating frequencies.
Electroencephalography (EEG) in Neuroscience: In EEG signals, certain brain wave patterns can be modulated by underlying rhythmic processes, such as the sleep-wake cycle. This can result in EEG waveforms where the primary rhythms are modulated, leading to complex patterns with distinct frequency bands.
Optical Communications: In optical fiber communications, a technique known as Wavelength Division Multiplexing (WDM) can be seen as modulating multiple light signals (each a periodic process) onto a single fiber. The combined signal exhibits a spectrum with distinct subbands, each corresponding to one of the modulated signals.
These examples illustrate the diversity of systems and contexts in which periodic impulse trains modulated by another periodic process can lead to complex waveforms with pronounced subbands, each reflecting the interplay between the modulating signals and the base periodic impulses.
The illustration above showcases vibration analysis in mechanical systems, depicting a complex waveform resulting from the modulation of a periodic impulse train by another periodic process. It includes a spectral analysis graph, highlighting the pronounced subbands within the waveform, and a representation of a mechanical system, such as a rotating shaft with an imbalance, along with the corresponding vibration signal being analyzed. This image is designed to be detailed and technical, suitable for educational purposes in the field of engineering.
More ChatGPT: The spectral domain representation below illustrates the result of a periodic impulse train, such as that from a rotating imbalance, being modulated by another vibrating periodic process. In this simulation, the periodic impulse train has a frequency of 50 Hz, and it is modulated by a lower frequency signal at 2 Hz.
The Fourier Transform of the modulated signal reveals pronounced subbands around the base frequency of the impulse train and its harmonics. These subbands are a direct consequence of the modulation process, where the amplitude of the impulse train is modulated by the lower frequency signal, leading to the creation of sidebands in the frequency domain. This graphical representation captures the complex interplay between the periodic impulse train and the modulating signal, resulting in a composite waveform characterized by these pronounced subbands.
This is a good recent citation : He, G., Ding, K., Li, W. and Li, Y. (2017), Frequency response model and mechanism for wind turbine planetary gear train vibration analysis. IET Renewable Power Generation, 11: 425-432. https://doi.org/10.1049/iet-rpg.2016.0236
Frequency components of complicated asymmetric modulation sidebands, existing in the vibration of the healthy planetary gear train, are prone to be erratically diagnosed as fault characteristics, which leads to difficulties in fault diagnosis of wind turbine planetary gearbox. The factors affecting the modulation sideband, i.e. the periodical time-varying transmission path and meshing force direction, are analysed. Considering both the meshing vibrations of the planet–ring and planet–sun gear pairs, a mathematical model was developed to analyse the planetary gear train’s vibration response. Simulation and experiments were conducted, and the mechanism of vibration modulation sidebands was revealed. The modulation sideband is not caused by the meshing vibration itself, but by the testing method that sensors are fixed on the ring gear or gearbox casing. The frequency components and amplitudes of the sidebands are determined by the tooth number of the ring gear and sun gear, the number of planet gears and their initial assembling phases. The asymmetric modulation sideband is mainly caused by the phase difference of the initial planets’ assembling phase.
Abstract: Frequency response model and mechanism for wind turbine planetary gear train vibration analysis
Is there a connection between the modulation of Laplace’s Tidal Equation (LTE) solutions and the highly nonlinear fits of neural networks?
“Neural tensor networks have been widely used in a large number of natural language processing tasks such as conversational sentiment analysis, named entity recognition and knowledge base completion. However, the mathematical explanation of neural tensor networks remains a challenging problem, due to the bilinear term. According to Taylor’s theorem, a kth order differentiable function can be approximated by a kth order Taylor polynomial around a given point. Therefore, we provide a mathematical explanation of neural tensor networks and also reveal the inner link between them and feedforward neural networks from the perspective of Taylor’s theorem. In addition, we unify two forms of neural tensor networks into a single framework and present factorization methods to make the neural tensor networks parameter-efficient. Experimental results bring some valuable insights into neural tensor networks.”
The connection is via Taylor’s series expansion whereby neural nets try to resolve tight inflection points that occur naturally in the numerical flow of potentially turbulent fluid dynamics.
Li, Wei, Luyao Zhu, and Erik Cambria. “Taylor’s theorem: A new perspective for neural tensor networks.” Knowledge-Based Systems 228 (2021): 107258.
Zhao, H., Chen, Y., Sun, D., Hu, Y., Liang, K., Mao, Y., … & Shao, H. “TaylorNet: A Taylor-Driven Generic Neural Architecture”. submitted to ICLR 2023
Also wavelets can maneuver tight inflections.
“In this study, the applicability of physics informed neural networks using wavelets as an activation function is discussed to solve non-linear differential equations. One of the prominent equations arising in fluid dynamics namely Blasius viscous flow problem is solved. A linear coupled differential equation, a non-linear coupled differential equation, and partial differential equations are also solved in order to demonstrate the method’s versatility. As the neural network’s optimum design is important and is problem-specific, the influence of some of the key factors on the model’s accuracy is also investigated. To confirm the approach’s efficacy, the outcomes of the suggested method were compared with those of the existing approaches. The suggested method was observed to be both efficient and accurate.”
Sorry to have to point this out, but it’s not my fault that geophysicists and climatologists can’t perform controlled experiments to test out various hypotheses. It’s not their fault either. It’s all nature’s decision to make gravitational forces so weak and planetary objects so massive to prevent anyone from scaling the effect to laboratory size to enable a carefully controlled experiment. One can always create roughly-equivalent emulations, such as a magnetic field experiment (described in the previous blog post) and validate a hypothesized behavior as a controlled lab experiment. Yet, I suspect that this would not get sufficient buy-in, as it’s not considered the actual real thing.
And that’s the dilemma. By the same token that analog emulators will not be trusted by geophysicists and climatologists, so too scientists from other disciplines will remain skeptical of untestable claims made by earth scientists. If nothing definitive comes out of a thought experiment that can’t be reproduced by others in a lab, they remain suspicious, as per their education and training.
It should therefore work both ways. As featured in the previous blog post, the model of the Chandler wobble forced by lunar torque needs to be treated fairly — either clearly debunked or considered as an alternative to the hazy consensus. ChatGPT remains open about the model, not the least bit swayed by colleagues or tribal bias. As the value of the Chandler wobble predicted by the lunar nodal model (432.7 days) is so close to the cited value of 433 days, as a bottom-line it should be difficult to ignore.
There are other indicators in the observational data to further substantiate this, see Chandler Wobble Forcing. It also makes sense in the context of the annual wobble.
As it stands, the lack of an experiment means a more equal footing for the alternatives, as they are all under equal amounts of suspicion.
Same goes for QBO. No controlled experiment is possible to test out the consensus QBO models, despite the fact that the Plumb and McEwan experiment is claimed to do just that. Sorry, but that experiment is not even close to the topology of a rotating sphere with a radial gravitational force operating on a gas. It also never predicted the QBO period. In contrast, the value of the QBO predicted by the lunar nodal model (28.4 months) is also too close to the cited value of 28 to 29 months to ignore. This also makes sense in the context of the semi-annual oscillation (SAO) located above the QBO .
Both the Chandler wobble and the QBO have the symmetry of a global wavenumber=0 phenomena so therefore only nodal cycles allowed — both for lunar and solar.
Next to ENSO. As with LOD modeling, this is not wavenumber=0 symmetry, as it must correspond to the longitude of a specific region. No controlled experiment is possible to test out the currently accepted models, premised as being triggered by wind shifts (an iffy cause vs. effect in any case). The mean value of the ENSO predicted by the tidal LOD-caibrated model (3.80 years modulated by 18.6 years) is too close to the cited value of 3.8 years with ~200 years of paleo and direct measurement to ignore.
In BLUE below is the LOD-calibrated tidal forcing, with linear amplification
In BLUE again below is a non-linear modulation of the tidal forcing according to the Laplace’s Tidal Equation solution, and trained on an early historical interval. This is something that a neural network should be able to do, as it excels at fitting to non-linear mappings that have a simple (i.e. low complexity) encoding — in this case it may be able to construct a Taylor series expansion of a sinusoidal modulating function.
The neural network’s ability to accurately represent a behavior is explained as a simplicity bias — a confounding aspect of machine learning tools such as ChatGPT and neural networks. The YouTube video below explains the counter-intuitive notion of how a NN with a deep set of possibilities tends to find the simplest solution and doing this without over-fitting the final mapping.
So that deep neural networks are claimed to have a built-in Occam’s Razor propensity, finding the most parsimonious input-output mappings when applied to training data. This is spot on with what I am doing with the LTE mapping, but bypassing the NN with a nonlinear sinusoidal modulation optimally fit on training data by a random search function.
I am tempted to try a NN on the ENSO training set as an experiment and see what it finds.
April 2, 2023
“I am tempted to try a NN on the ENSO training set as an experiment and see what it finds.”
Nonlinear aspects plays a major role in the understanding of fluid flows. The distinctive fact that in nonlinearproblems cause and effect are not proportional opens up thepossibility that a small variation in an input quantity causes aconsiderable change in the response of the system. Often thistype of complication causes nonlinear problems to eludeexact treatment. A good illustration of this feature is the factthat there is only one known explicit exact solution of the(nonlinear) governing equations for periodic two-dimensional traveling gravity water waves. This solution was firstfound in a homogeneous fluid by Gerstner
These are trochoidal waves
Even within the context of gravity waves explored in the references mentioned above, a vertical wall is not allowable. This drawback is of special relevance in a geophysical context since [cf. Fedorov and Brown, 2009] the Equator works like a natural boundary and equatorially trapped waves, eastward propagating and symmetric about the Equator, are known to exist. By the 1980s, the scientific community came to realize that these waves are one of the key factors in explaining the El Niño phenomenon (see also the discussion in Cushman-Roisin and Beckers [2011]).
Experimenting with linking to slide presentations instead of a trad blog post. The PDF linked below is an eye-opener as the NINO34 fit is the most parsimonious ever, at the expense of a higher LTE modulation (explained here). The cross-validation involves far fewer tidal factors than dealt with earlier, the two factors used (Mf and Mm tidal factors) rivaling the one factor used in QBO (described here).
Following up from this post, there is a recent sequence of articles in an AGU journal on Water Resources Research under the heading: “Debates: Does Information Theory Provide a New Paradigm for Earth Science?”
By anticipating all these ideas, you can find plenty of examples and derivations (with many centered on the ideas of Maximum Entropy) in our book Mathematical Geoenergy.
Here is an excerpt from the “Emerging concepts” entry, which indirectly addresses negative entropy:
“While dynamical system theories have a long history in mathematics and physics and diverse applications to the hydrological sciences (e.g., Sangoyomi et al., 1996; Sivakumar, 2000; Rodriguez-Iturbe et al., 1989, 1991), their treatment of information has remained probabilistic akin to what is done in classical thermodynamics and statistics. In fact, the dynamical system theories treated entropy production as exponential uncertainty growth associated with stochastic perturbation of a deterministic system along unstable directions (where neighboring states grow exponentially apart), a notion linked to deterministic chaos. Therefore, while the kinematic geometry of a system was deemed deterministic, entropy (and information) remained inherently probabilistic. This led to the misconception that entropy could only exist in stochastically perturbed systems but not in deterministic systems without such perturbations, thereby violating the physical thermodynamic fact that entropy is being produced in nature irrespective of how we model it.
In that sense, classical dynamical system theories and their treatments of entropy and information were essentially the same as those in classical statistical mechanics. Therefore, the vast literature on dynamical systems, including applications to the Earth sciences, was never able to address information in ways going beyond the classical probabilistic paradigm.”
That is, there are likely many earth system behaviors that are highly ordered, but the complexity and non-linearity of their mechanisms makes them appear stochastic or chaotic (high positive entropy) yet the reality is that they are just a complicated deterministic model (negative entropy). We just aren’t looking hard enough to discover the underlying patterns on most of this stuff.
“Science and data compression have the same objective: discovery of patterns in (observed) data, in order to describe them in a compact form. In the case of science, we call this process of compression “explaining observed data.” The proposed or resulting compact form is often referred to as “hypothesis,” “theory,” or “law,” which can then be used to predict new observations. There is a strong parallel between the scientific method and the theory behind data compression. The field of algorithmic information theory (AIT) defines the complexity of data as its information content. This is formalized as the size (file length in bits) of its minimal description in the form of the shortest computer program that can produce the data. Although complexity can have many different meanings in different contexts (Gell-Mann, 1995), the AIT definition is particularly useful for quantifying parsimony of models and its role in science. “
Parsimony of models is a measure of negative entropy
The objective is to apply negative entropy to find an optimal solution to a deterministically ordered pattern. To start, let us contrast the behavior of autonomous vs non-autonomous differential equations. One way to think about the distinction is that the transfer function for non-autonomous only depends on the presenting input. Thus, it acts like an op-amp with infinite bandwidth. Or below saturation it gives perfectly linear amplification, so that as shown on the graph to the right, the x-axis input produces an amplified y-axis output as long as the input is within reasonable limits.
Something I learned early on in my research career is that complicated frequency spectra can be generated from simple repeating structures. Consider the spatial frequency spectra produced as a diffraction pattern produced from a crystal lattice. Below is a reflected electron diffraction pattern of a reconstructed hexagonally reconstructed surface of a silicon (Si) single crystal with a lead (Pb) adlayer ( (a) and (b) are different alignments of the beam direction with respect to the lattice). Suffice to say, there is enough information in the patterns to be able to reverse engineer the structure of the surface as (c).
Now consider the ENSO pattern. At first glance, neither the time-series signal nor the Fourier series power spectra appear to be produced by anything periodically regular. Even so, let’s assume that the underlying pattern is tidally regular, being comprised of the expected fortnightly 13.66 day tropical/synodic cycle and the monthly 27.55 day anomalistic cycle synchronized by an annual impulse. Then the forcing power spectrum of f(t) looks like the RED trace on the left-side of the figure below, F(ω). Clearly that is not enough of a frequency spectra (a few delta spikes) necessary to make up the empirically calculated Fourier series for the ENSO data comprising ~40 intricately placed peaks between 0 and 1 cycles/year in BLUE.
click to expand
Yet, if we modulate that with an Laplace’s Tidal Equation solution functional g(f(t)) that has a G(ω) as in the yellow inset above — a cyclic modulation of amplitudes where g(x) is described by two distinct sine-waves — then the complete ENSO spectra is fleshed out in BLACK in the figure above. The effective g(x) is shown in the figure below, where a slower modulation is superimposed over a faster modulation.
So essentially what this is suggesting is that a few tidal factors modulated by two sinusoids produces enough spectral detail to easily account for the ~40 peaks in the ENSO power spectra. It can do this because a modulating sinusoid is an efficient harmonics and cross-harmonics generator, as the Taylor’s series of a sinusoid contains an effectively infinite number of power terms.
To see this process in action, consider the following three figures, which features a slider that allows one to get an intuitive feel for how the LTE modulation adds richness via harmonics in the power spectra.
Start with a mild LTE modulation and start to increase it as in the figure below. A few harmonics begin to emerge as satellites surrounding the forcing harmonics in RED.
drag slider right for less modulation and to the left for more modulation
2. Next, increase the LTE modulation so that it models the slower sinusoid — more harmonics emerge
3. Then add the faster sinusoid, to fully populate the empirically observed ENSO spectral peaks (and matching the time series).
It appears as if by magic, but this is the power of non-linear harmonic generation. Note that the peak labeled AB amongst others is derived from the original A and B as complicated satellite-cross terms, which can be accounted for by expanding all of the terms in the Taylor’s series of the sinusoids. This can be done with some difficulty, or left as is when doing the fit via solver software.
To complete the circle, it’s likely that being exposed to mind-blowing Fourier series early on makes Fourier analysis of climate data less intimidating, as one can apply all the tricks-of-the-trade, which, alas, are considered routine in other disciplines.
GeoEnergy Math 