arXiv subjects

Tomoro Mochida

Publications and source records attributed to Tomoro Mochida.

4 recordsLinked to original sources

arXiv2026

Cohomology for solutions of polygon equations

Polygon equations form a family of equations generalizing the pentagon equation. In this paper, we construct semi-simplicial sets of permitted colorings associated with set-theoretic solutions of polygon equations and use them to define the corresponding (co)homology groups. We investigate several properties of these groups and establish an equivalence of categories between set-theoretic solutions of polygon equations and higher Segal semi-simplicial sets satisfying certain conditions. As a special case, our result recovers the correspondence between bijective set-theoretic solutions of the pentagon equation and $2$-Segal semi-simplicial sets proved by Dyckerhoff--Kapranov.

math-ph↗

arXiv2025

Constructing Solutions of Simplex Equations from Polygon Equations

We study polygon equations and their connections to simplex equations, which generalize the pentagon and Yang-Baxter equations, respectively. First, we show that certain ''commutative'' pairs of solutions of (dual) polygon equations give rise to solutions of higher-order polygon equations. Next, we define an explicit compatibility condition between solutions of the $n$-gon and dual $n$-gon equations and use it to construct solutions of the $(n-2)$- and $(n-1)$-simplex equations. This extends earlier work by Kashaev-Sergeev and Dimakis-M\"uller-Hoissen.

math-ph↗

arXiv2024

Invariants of flat connections on 4-manifolds from Hopf group-algebras

For a given group $G$, we construct an invariant of flat $G$-connections on 4-manifolds from a finite type involutory quasitriangular Hopf $G$-algebra. Hopf $G$-algebras are generalizations of Hopf algebras, equipped with gradings by $G$. In our construction, we color the dotted components of a Kirby diagram with elements of $G$ and employ the Hennings-type procedure. When $G$ is finite, we also define an invariant of 4-manifolds by summing the invariants over all flat $G$-connections.

math.GT↗

arXiv2024

Two Online Map Matching Algorithms Based on Analytic Hierarchy Process and Fuzzy Logic

Our aim of this paper is to develop new map matching algorithms and to improve upon previous work. We address two key approaches: Analytic Hierarchy Process (AHP) map matching and fuzzy logic map matching. AHP is a decision-making method that combines mathematical analysis with human judgment, and fuzzy logic is an approach to computing based on the degree of truth and aims at modeling the imprecise modes of reasoning from 0 to 1 rather than the usual boolean logic. Of these algorithms, the way of our applying AHP to map matching is newly developed in this paper, meanwhile, our application of fuzzy logic to map matching is mostly the same as existing research except for some small changes. Because of the common characteristic that both methods are designed to handle imprecise information and simplicity for implementation, we decided to use these methods.

cs.CG↗