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| ai.viXra.org > Functions and Analysis > 2606 |
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[2] ai.viXra.org:2606.0013 [pdf] submitted on 2026-06-04 20:16:22
Authors: Richard Andrew Holland
Comments: 30 Pages.
We present an efficient and consistent method for discerning preferable and generating functions of data using filtered spectral fingerprints. Spectral analysis of functions and data is hardly new. For example, with Fourier Transforms one can classify and identify functions using coefficients. The problem is that statistical, domain and systematic error noise is baked into the spectrum of coefficients. For over two centuries Legendre polynomials have been used for spectra, however, we are unaware of prior use of this approach. We create a hash library of function spectra to quickly find preferable and/or generating functions. Software tools (Patents Pending) have been developed utilizing this methodology, including one for finding AI/ML Neuron Activation functions. Even for noisy and instrument biased data, researchers and machine optimizers can better discern preferential and underlying functions that generate noisy data. Preliminary AI/ML results are also presented.
Category: Functions and Analysis
[1] ai.viXra.org:2606.0009 [pdf] submitted on 2026-06-03 20:02:38
Authors: Haoxiang Yu
Comments: 11 Pages. See https://github.com/yhx-12243/Sidon3 for formalization
We determine the exact value of the Sidon constant of the four-element set {0,1,2,3} to be 5/3. The lower bound was established by Neuwirth via an explicit family of trigonometric polynomials; we prove the matching upper bound. Our proof introduces a new method: given a point on the unit circle we construct a cubic self-inversive polynomial whose three roots lie on the unit circle, extract positive real weights from these roots, and use a weighted square-sum identity to obtain the sharp estimate. The entire proof is formalized in Lean 4 using mathlib.
Category: Functions and Analysis