| viXra.org > Functions and Analysis > 2507 |
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| viXra.org > Functions and Analysis > 2507 |
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[3] viXra:2507.0216 [pdf] replaced on 2025-08-04 04:26:38
Authors: Shalom Keshet
Comments: 33 Pages. Author’s Note: The language of the manuscript has been updated for improved clarity and readability. All remaining content, including mathematical derivations and structure, remains unchanged.
We present a unified algebraic framework for representing and transforming arbitrary piecewise-defined functions into smooth, differentiable expressions suitable for analysis and optimization. We introduce the Kronecker naught operator, an analytic analog of the Kronecker delta, to encode indicator-style discontinuities via Fourier-series and trigonometric membership conditions. Our approach systematically replaces hard conditional branches with "soft" approximations, parameterized smooth maxima, Heaviside steps, and differentiable logical operators, that converge to the original piecewise form in the sharp-limit. We further extend these techniques to number-theoretic and combinatorial domains, showing how divisibility and integer-membership tests can be expressed through sine-squared filters and gamma-function identities. We also introduce the core concept of a conditional function, which we will denote with C(S) for some mathematical statement S, which will be key to converting these piecewise functions into a more straightforward entity. Finally, we apply our algebraic toolkit to a continuous reformulation of the 3x + 1 (Collatz) mapping, yielding a novel smooth dynamical system that retains the integer-orbital structure while admitting gradient-based analysis. Throughout, we illustrate key constructions with explicit formulas, discuss convergence properties, and highlight connections to modern machine-learning architectures that rely on differentiable decision boundaries. This work lays a foundation for both theoretical investigations of piecewise phenomena and practical implementations in differentiable programming environments.
Category: Functions and Analysis
[2] viXra:2507.0192 [pdf] submitted on 2025-07-26 19:21:13
Authors: Alexis Zaganidis
Comments: 12 Pages.
The first part of the present article is about advanced trigonometry formulas involving a rotating sphere and a source light infinitely far way. The second part of the present article is about two advanced conjectures on the global minimum of functions of class $C^2$ defined over the real coordinate space $mathbb {R}^n$. The third part of the present article is about an advanced definition of the weak asymptomatic limit of functions of a single strictly positive real variable by applying recursively a cutoff function.
Category: Functions and Analysis
[1] viXra:2507.0031 [pdf] submitted on 2025-07-04 14:31:20
Authors: Urs Frauenfelder, Joa Weber
Comments: 81 pages, 8 figures
In Floer theory one has to deal with two-level manifolds like for instance the space of $W^{2,2}$ loops and the space of $W^{1,2}$ loops. Gradient flow lines in Floer theory are then trajectories in a two-level manifold. Inspired by our endeavor to find a general setup to construct Floer homology we therefore address in this paper the question if the space of paths on a two-level manifold has itself the structure of a Hilbert manifold. In view of the two topologies on a two-level manifold it is unclear how to define the exponential map on a general two-level manifold. We therefore study a different approach how to define charts on path spaces of two-level manifolds. To make this approach work we need an additional structure on a two-level manifold which we refer to as emph{tameness}. We introduce the notion of tame maps and show that the composition of tame is tame again. Therefore it makes sense to introduce the notion of a tame two-level manifold. The main result of this paper shows that the path spaces on tame two-level manifolds have the structure of a Hilbert manifold.
Category: Functions and Analysis
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