Math Papers
Structural boundaries, partitions and resonance of radical-prime multiplicative sequences
We look at the structural properties, algebraic boundaries, and asymptotic behavior of integer sequences whose terms are defined by a resonance between an integer's radical kernel and its prime-factor-counting functions. The main object of study is A397221 (k = rad(k) * omega(k)) and A396594 (k = rad(k) * BigOmega(k)); we prove that the only powerful element of the latter is 4 and that its only squarefree elements are the primes. We then look at the non-prime elements and show they form a proper subset of the "squaremid" numbers (A332785). From there we build a partition of the squaremid space into three disjoint sets (A397499, A397500, and A396367) and prove they don't overlap.
Trying to eradicate Mandelbrot Bulbous via Maps and novel fractals on complex space
Standard polynomial maps of the form $z_{n+1} = z_n^d + c$ invariably manifest structural cardioids or localized bulbous branch points within their parameter spaces. This paper explores alternative algebraic architectures that systematically suppress standard Mandelbrot-like morphology. We introduce three novel families of complex dynamical systems: parameter-variable transcendental tetration, rational maps bound by invariant imaginary poles, and non-holomorphic parameter-warped systems. Numerical experiments verify that these formulations disrupt classical boundary accumulation, yielding highly geometric, non-biomorphic filamentary networks and sharp phase-space fractures.
The convergence of SHA-256 digit-sum orbits to fixed-point attractors
An analysis of discrete dynamical systems generated by iterating the composition of the SHA-256 cryptographic hash function and the base-10 digit-sum function, demonstrating empirical convergence toward fixed-point attractors.
A proposal for a Fractal Catalog
This article proposes the creation of an Online Encyclopedia of Fractals (OEF), a comprehensive, standardized repository for cataloging the infinite diversity of fractal geometries. The OEF would utilize a novel, line-oriented domain-specific language called Fracta to encode fractal generation algorithms in a compact, human-readable format. The article also addresses the challenges of duplicate detection and mathematical mimicry in fractal classification, proposing automated visual fingerprinting and empirical dimension estimation techniques to ensure a clean, non-redundant database.
Symbolic Synthesis of Eulers Number and the Golden Ratio
This paper analyzes the structural configurations generated by symbolic regression driven by genetic algorithms (GAs) when tasked with approximating or defining fundamental mathematical constants ($e$ and $\Phi$). Unbound by human design heuristics—such as the classical preference for visual symmetry and algebraic parsimony—the evolutionary process yields unconventional functional architectures. We examine the mechanics of asymptotic convergence, the automated leveraging of removable singularities, and the algebraic invariance of the evolved continued fractions.
Fracta Language Specification: A Deterministic Syntax for Fractal
This document defines the formal grammar, lexical structure, and runtime execution semantics of Fracta v1.0, a minimalistic, line-oriented domain-specific language (DSL) designed for the generation of recursive geometric topologies. Fracta maps string-rewriting Lindenmayer Systems (L-Systems) directly into deterministic vector trajectories via a stack-based turtle graphics state machine. The language optimizes syntactic overhead to achieve low cognitive friction and high parse efficiency in resource-constrained or distributed environments.
Online Encyclopedia of Fractals (OEF) Standard
This document specifies the Data Schema, Meta-syntax, and Execution Core for the Online Encyclopedia of Fractals (OEF) alongside its accompanying Domain-Specific Language, Fracta. OEF provides a plaintext, peer-submittable registry for mathematical fractals, architectural structures, and chaotic systems. It addresses the structural equivalence challenge (where disparate algorithmic generators produce identical geometric attractors) by indexing topologies via an automated box-counting Hausdorff dimension algorithm combined with a discrete, coordinate-normalized Jaccard intersection engine. This enables algorithmic duplicate prevention, deterministic cataloging, and the automatic classification of mathematical anomalies.
GES: a Graph Encyclopedia
A data schema, canonicalization protocol, serialization format, and query language for the Graph Encyclopedia Standard (GES). GES provides a plaintext, peer-submittable registry of finite graphs and combinatorial structures.