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Previous months:
2025 - 2504(3) - 2505(1)
2026 - 2601(1) - 2602(2) - 2604(1) - 2605(1) - 2606(2)
Any replacements are listed farther down
[11] 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
[10] 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
[9] ai.viXra.org:2605.0076 [pdf] submitted on 2026-05-31 01:57:27
Authors: Dagony P. Rony, Blank T. Blazer
Comments: 13 Pages.
On the Algebraic Constitution of the Sahur: A Complex-Analytic Theory of the n-Tung Sahur Function, with Consequences for the Ontology of Real and Imaginary Drum-EntitiesWe develop a rigorous algebraic and complex-analytic theory of the Sahur, the wooden percussive night-entity of the Indonesian Tung Tung Tung Sahur tradition as transmitted through contemporary short-form video culture. Taking as our sole non-trivial postulate the identity TTTS = i — the assertion that the canonical three-tung sahur is the imaginary unit — together with the colloquial "triple-T" relation 3T = TTTS, we solve the resulting system to obtain T = i/3 and S = -27. From these constants we construct the n-tung sahur function s(n) = Tu207f·S = -27(i/3)u207f, extend it to an entire function s: ℂ → ℂ, and study its analytic, dynamical, and spectral structure. Our central result (the Parity Manifestation Theorem) establishes that s(n) is purely real precisely at even integer tung-counts and purely imaginary at odd ones; since the empirically observed three-tung sahur is odd, the historically attested entity is necessarily imaginary, yet the theorem simultaneously guarantees the existence of genuinely real sahurs, resolving in the affirmative the long-standing question of whether a sahur may be real in real life. We locate the Sahur of Maximal Manifestation at the fractional tung-count n* = 1 + (2/π)·arctan(π/(2 ln 3)) ≈ 1.6115, characterise the orbit {s(n)} as a logarithmic-spiral attractor of the contraction-rotation z ↦ (i/3)z, treat the multivaluedness of fractional tung-counts via the associated Riemann surface, and discuss physical interpretations through the lens of complex resonance and damped oscillation. Numerous graphs, footnotes, and one dissenting appendix are provided.
Category: Functions and Analysis
[8] ai.viXra.org:2604.0104 [pdf] submitted on 2026-04-30 17:29:15
Authors: Yahor Hous
Comments: 13 Pages.
We present a two-part analysis of the Type I blow-up scenario and the Liouville property for the three-dimensional incompressible Navier—Stokes equations.
Category: Functions and Analysis
[7] ai.viXra.org:2602.0067 [pdf] submitted on 2026-02-14 01:33:55
Authors: Cornelius Moore
Comments: 24 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
Modern theoretical physics employs distinct mathematical formalisms—Lagrangian mechanics, Hamiltonian dynamics, quantum amplitudes, statistical ensembles, field-theoretic path integrals—that, while empirically successful, lack a unified structural foundation. We present the Universal Mathematical System (UMS), a vari-ational—probabilistic framework in which standard physical theories arise as limiting cases, projections (marginalizations), or constrained reductions of a single maximum-entropy measure over configuration spaces. The framework is built on an exponential-family measure µ[C] = Z−1 exp(−Φ[C]), where Φ is a constraint functional encoding physical laws. We show under standard regularity assumptionshow classical mechanics emerges via the Laplace principle (β →∞), statistical mechanics is directly identified with the framework, and quantum mechanics corresponds to complex-weighted measures under standard path-integral formalism. Additionally, we formalize five distinct algebraic structures—quantity (Rn), growth (semigroups), information (entropy), phase (U(1)), and ratio (R+)—clarifying thatdifferent physical questions inhabit different mathematical domains and that confusion arises from naive cross-domain interpretation. The framework is intended as structural unification of existing formalisms rather than a proposal of novel fundamental ontology or new empirical predictions. We include a proof of a coarse-graining monotonicity theorem using the data-processing inequality, provide explicitreduction pipelines, and discuss extensions to chemistry, biology, neuroscience, and computation.
Category: Functions and Analysis
[6] ai.viXra.org:2602.0010 [pdf] submitted on 2026-02-03 19:55:50
Authors: Mohammad Saeed Alnatour
Comments: 172 Pages. (Note by ai.viXra.org Admin: For the last time, author name is required in the article after article title and please cite listed scientific references)
Divergent series and singular integrals arise naturally in analysis, geometry, and theoretical physics, yet their standard treatment relies on analytic continuation or limit-based regularization. While these methods successfully assign finite values, they necessarily suppress information about how infinity is traversed. This work proposes a structural framework—Interconnected Infinities Giant Sphere Space (IIGSS), together with an intrinsic regulator, the Discrete Laplace Regulator (DLR), in which divergence is treated as a boundary phenomenon rather than a failure of summation.DLR operates directly on discrete sequences by introducing controlled exponential damping and expanding the resulting kernel at a well-defined infinite-traversal gate. Divergence appears explicitly as algebraic pole terms or logarithmic singularities in the gate expansion, encoding growth class and traversal density, while a pole-invariant constant—the Convergence Momentum (CM)—emerges as a finite structural quantity. Valuation is performed exclusively through gate expansion followed by pole removal, without index shifting, limit evaluation, or analytic continuation.Within this framework, classical zeta and Dirichlet regularizations are recovered as special projections under standard traversal, while traversal-sensitive features—such as zero insertion, spacing modulation, and phase structure—remain distinguishable. The framework accommodates finite-gate and oscillatory sequences and clarifies the limitations of reconstruction from regularized values alone. In physical applications, CM functions as a retained boundary invariant: when applied to spectral mode sums, such as those appearing in the Casimir effect, the regulator preserves observable finite quantities while rendering the underlying divergence structure explicit. DLR thus provides a higher-resolution language for infinity, preserving established results while exposing structural information necessarily omitted by classical methods.
Category: Functions and Analysis
[5] ai.viXra.org:2601.0014 [pdf] submitted on 2026-01-05 18:19:37
Authors: Boudjema Abderraouf
Comments: 12 Pages. (Note by ai.viXra.org Admin: Please cite and list scientific references)
The Growth Depth Index (GDI) system provides a comprehensive framework for classifying asymptotic function growth. By capturing both structural hierarchy (depth) and quantitative parameters (rate, exponent coefficient, multiplicative constant), it enables precise comparison of functions across all common asymptotic classes. The key innovation is the preservation of multiplicative constants throughcareful parameter extraction, making the system suitable for implementation in function comparison tools and asymptotic analysis.
Category: Functions and Analysis
[4] ai.viXra.org:2505.0011 [pdf] submitted on 2025-05-02 23:50:29
Authors: Chang Hee Kim
Comments: 8 Pages. Licensed under CC BY-NC-ND 4.0
This paper presents a new foundational insight into the Fast Fourier Transform (FFT) algorithm: mixed-radix decomposition is not an optional design, but an inherent structural property of FFT itself. We demonstrate that any power series f(x), when decomposed into modulo-s subseries for any integer s ≥ 2, naturally aligns with radix-s FFT computation — regardless of whether s is prime. This unification renders the need for specialized mixed-radix FFT frameworks obsolete. Furthermore, we show that this structure enables seamless extraction of Puiseux series coefficients from a function, including those involving fractional powers and branch cuts. The FFT not only resolves monodromy behavior computationally, but also simplifies the treatment of multivalued functions across Riemann surfaces without symbolic intervention. This paper is a direct sequel to the author’s previous work, Sampling on the RiemannSurface: A Natural Resolution of Branch Cuts in Puiseux Series, and the first foundational paper, A Unified Computational Framework Unifying Taylor-Laurent, Puiseux, Fourier Series, and theFFT Algorithm.
Category: Functions and Analysis
[3] ai.viXra.org:2504.0104 [pdf] submitted on 2025-04-26 17:31:39
Authors: Chang Hee Kim
Comments: 8 Pages. (Note by viXra Admin: Please cite and list scientific references)
This paper presents a novel yet profoundly natural approach to handling branch cuts in multivalued complex functions, especially Puiseux series, by sampling directly on the Riemann surface. Rather than imposing artificial branch cuts, we explore how FFT-based sampling across monodromy inherently respects the geometry of multivalued functions. The method aligns with the deep structure of Riemann surfaces and offers a unified computational framework for extracting coefficients in power series, including fractional exponents.
Category: Functions and Analysis
[2] ai.viXra.org:2504.0062 [pdf] submitted on 2025-04-19 19:38:41
Authors: Robert L. Jackson
Comments: 16 Pages. This article applies GFT for deriving solutions to inhomogeneous NPDEs.
This article emphasizes the fine-tuning step of the Generating Function Technique (GFT), a crucial component enhancing solution accuracy and computational efficiency for nonlinear partial differential equations (NPDEs). Unlike other methods, such as the Simplest Equation Method (SEM) and the G’/G-expansion method, the fine-tuning step within GFT systematically optimizes the solution series. This paper demonstrates fine-tuning’s impact through detailed applications to the inhomogeneous Bateman-Burgers and Good Boussinesq equations, elucidating its capability in generating superior analytical solutions.
Category: Functions and Analysis
[1] ai.viXra.org:2504.0027 [pdf] submitted on 2025-04-10 19:44:18
Authors: Chang Hee Kim
Comments: 30 Pages. English Includes C++ code in the appendix for demonstration Copyright License: Creative Commons Attribution-NonCommercial 4.0 International License
This paper proposes a unified framework for understanding and computing Taylor-Laurent Series, Puiseux Series, Fourier Series, and the Fast Fourier Transform (FFT) algorithm. By examining the connections between these seemingly disparate mathematical concepts, we show that they are not distinct, but manifestations of a single computational principle. We explore their shared foundation in series expansion, their computational embodiment via linear systems, and the central role of FFT as the universal engine.
Category: Functions and Analysis
[4] ai.viXra.org:2504.0062 [pdf] replaced on 2025-08-01 00:04:03
Authors: Robert L. Jackson
Comments: 9 Pages. Obtain zip files via https://notebookarchive.org/2025-07-9qxwu8n. Contact information is rljacksonmd@gmail.com
This article emphasizes the fine-tuning step of the Generating Function Technique (GFT), a crucial component that enhances solution accuracy and computational efficiency for nonlinear partial differential equations (NPDEs). Unlike other methods, such as the Simplest Equation Method, G’/G-Expansion Method, Adomain Decomposition, and Homotopy Perturbative Method, the fine-tuning step within GFT systematically optimizes the solution series. This paper demonstrates the impact of fine-tuning through detailed applications to inhomogeneous NPDEs, elucidating its capability in generating superior analytical solutions.
Category: Functions and Analysis
[3] ai.viXra.org:2504.0062 [pdf] replaced on 2025-07-22 18:41:52
Authors: Robert L. Jackson
Comments: 30 Pages. I will place more examples on the Wolfram Cloud Notebook Archive soon. Contact information is rljacksonmd@gmail.com
This article emphasizes the fine-tuning step of the Generating Function Technique (GFT), a crucial component that enhances solution accuracy and computational efficiency for nonlinear partial differential equations (NPDEs). Unlike other methods, such as the Simplest Equation Method, G’/G-Expansion Method, Adomain Decomposition, and Homotopy Perturbative Method, the fine-tuning step within GFT systematically optimizes the solution series. This paper demonstrates the impact of fine-tuning through detailed applications to inhomogeneous NPDEs, elucidating its capability in generating superior analytical solutions.
Category: Functions and Analysis
[2] ai.viXra.org:2504.0062 [pdf] replaced on 2025-06-18 17:24:31
Authors: Robert L. Jackson
Comments: 16 Pages. contact at rljacksonmd@gmail.com
This article emphasizes the fine-tuning step of the Generating Function Technique (GFT), a crucial component enhancing solution accuracy and computational efficiency for nonlinear partial differential equations (NPDEs). Unlike other methods, such as the Simplest Equation Method (SEM) and the G’/G-expansion method, the fine tuning step within GFT systematically optimizes the solution series. This paper demonstrates fine-tuning’s impact through detailed applications to the inhomogeneous Bateman-Burgers and Boussinesq equations, elucidating its capability in generating superior analytical solutions.
Category: Functions and Analysis
[1] ai.viXra.org:2504.0062 [pdf] replaced on 2025-04-25 14:41:13
Authors: Robert L. Jackson
Comments: 12 Pages. contact at rljacksonmd@gmail.com
This article emphasizes the fine-tuning step of the Generating Function Technique (GFT), a crucial component enhancing solution accuracy and computational efficiency for nonlinear partial differential equations (NPDEs). Unlike other methods, such as the Simplest Equation Method (SEM) and the G’/G-expansion method, the fine tuning step within GFT systematically optimizes the solution series. This paper demonstrates fine-tuning’s impact through detailed applications to the inhomogeneous Bateman-Burgers and Boussinesq equations, elucidating its capability in generating superior analytical solutions.
Category: Functions and Analysis