159 papers
This collection explores the fascinating world of mathematical structures that connect directly to group theory, often abbreviated as "Gm" in specialized literature. These preprints delve into the abstract rules governing symmetry and transformation, offering insights that ripple through physics, cryptography, and computer science. By breaking down complex proofs and theoretical models, these works reveal how pure mathematics builds the foundation for understanding the patterns of our universe.
Every new preprint in this category is sourced directly from arXiv, the leading hub for scientific research. At Gist.Science, we process each submission to provide both a clear, plain-language overview and a detailed technical summary, ensuring that groundbreaking discoveries in group mathematics are accessible to everyone, regardless of their background. Below are the latest papers in this dynamic field, ready for you to explore.
This paper investigates the open problem of the irrationality of by establishing exact arithmetic equivalences for its hypothetical rationality and conducting a comprehensive "no-go" audit of low-complexity Apéry-type proof mechanisms, ultimately demonstrating that within tested families, analytic smallness is consistently obstructed by denominator growth and continued-fraction shadows.
This paper establishes classical exponential bounds for p-generalized circular and hyperbolic functions, thereby extending existing results for their classical counterparts.
This paper establishes necessary and sufficient conditions for the group inverse of a specific class of singular, irreducible, symmetric -matrices—motivated by recoverable complete networks—to retain the -matrix property, utilizing both matrix-theoretic methods and network potential theory to construct such matrices and deepen the link between -matrix theory and network analysis.
This paper presents a machine-assisted approach to automated conjecture discovery using the HypothesiX agent to explore twin prime distributions while introducing a Mahalanobis distance-based benchmark to quantify the non-triviality of generated conjectures and localize verification errors in autoformalizers.
This paper presents new lower bounds for the degree/diameter problem, specifically and , achieved through a novel discovery process where the author interacted directly with a standard web-accessible LLM to construct explicit graphs without using custom agent frameworks or pre-defined search strategies.
This paper establishes a lattice-theoretic framework for convexity that separates domain segment structures from codomain lattice structures to demonstrate that symmetric hesitant fuzzy convexity cannot be fully reconstructed by any finite family of scalar observables, thereby revealing the limitations of scalar reductions in preserving intrinsic order-theoretic information.
This paper bridges combinatorial geometry and music theory to solve the topological challenge of embedding Neo-Riemannian harmony into 19-TET by proving that 32 of 36 harmonic connections can be preserved through a unique canonical configuration, while simultaneously designing a novel split-key keyboard to make this microtonal system physically playable.
This paper presents a simplified proof of Laplace's formula using elementary nonstandard analysis to reduce the complexity of limits and clarify the underlying logic compared to classical calculus.
This paper develops a spectral theory for scale-invariant operators on the multiplicative half-line using Mellin transforms to demonstrate that geometric and spectral exponents are fundamentally decoupled, providing a precise mathematical characterization of multicriticality where their inequality signals multiple independent scaling dimensions.
This paper introduces new variants of the $abc$-conjecture based on doubly geometric mean quality metrics, establishing asymptotic results and sharp phase transitions for high-quality triples while developing sub-linear algorithms to efficiently identify them.