arXiv:2607. 24847v1 Announce Type: cross Abstract: We introduce an extremal invariant associated with Chowla-type order conditions in finite groups.
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arXiv:2607. 23732v1 Announce Type: cross Abstract: In Question~3.
arXiv:2609. 03246v1 Announce Type: cross Abstract: Smoothness of a function on the real line is reflected in the decay of its Fourier transform, which suggests that smoothness of a function in $L^2(G)$ for a group $G$ should mean concentration of the Fourier coefficients at low frequency.
arXiv:2608. 07275v1 Announce Type: cross Abstract: A group is an E-group if every element commutes with each of its endomorphic images.
arXiv:2609. 17578v1 Announce Type: cross Abstract: Let $a_1,\ldots,a_m$ be distinct elements of a field $K$, and let $Z(X)=\prod_{i=1}^m (X-a_i)$.
arXiv:2607. 23004v1 Announce Type: cross Abstract: Let $t(N)$ be the largest $t$ for which there exist distinct sets $A_1,\dots,A_t \subseteq \{1,\dots,N\}$ such that $A_i \cap A_j$ is a nonempty arithmetic progression for all $i \neq j$ (Erdos Problem #272).
The paper investigates exact factorization and canonical presentations of languages relative to a fixed finite‑monoid observation. It shows that unique factorization does not guarantee a finite relative presentation property (FRP) by presenting a 36‑element quotient with infinite valid prime‑return rules, and introduces the stronger finite‑state relative presentation property (FSRP). The authors further define prime‑target left‑division determinism (PTLD), prove its implications for factorization and rule bounds, and provide efficient learning algorithms for the canonical PTLD presentation and FSRP controller.