インサイト - Mathematics

### Interpreting the Lattice of Dyck Paths as a Lattice of Subcategories Using ω-Torsion Pairs

This note demonstrates a novel interpretation of the lattice of Dyck paths as a lattice of subcategories, specifically ω-torsion pairs, within the context of representation theory of finite-dimensional algebras.

### The Minimum Number of Maximal Independent Sets in Graphs with Given Order and Independence Number

This research paper investigates the minimum number of maximal independent sets (MIS) in trees and unicyclic graphs when both the order (number of vertices) and independence number (size of the largest independent set) are fixed.

### Categorifying Valuative Invariants of Polyhedra and Matroids via Categorical Convolution

This paper introduces the concept of "categorical valuative invariants" for polyhedra and matroids, which elevates traditional numerical invariants to exact sequences in additive categories, offering a deeper understanding of valuativity and enabling computations that respect matroid symmetries.

### On the Two-Color Rado Number for a Specific Linear Equation

This research paper investigates the two-color Rado numbers for a specific class of linear equations, providing exact values in some cases and upper and lower bounds in others, thereby generalizing previous findings.

### Improved Upper Bounds for Generalized Ramsey Numbers in the Non-Integral Regime

This paper presents an improved upper bound for generalized Ramsey numbers in the non-integral regime, achieved by utilizing the Forbidden Submatching Method and introducing a novel potential function to define forbidden sub-configurations.

### The Characterization of Sobriety in L-Convex Spaces

This research paper investigates the concept of sobriety in the context of L-convex spaces, a generalization of traditional convex spaces to a fuzzy setting. The authors establish a characterization for the sobrification of L-convex spaces and demonstrate that sober L-convex spaces are precisely the strictly injective objects in the category of S0 L-convex spaces.

### On z-Ideals and z-Closure Operations of Semirings, I

This paper introduces the concept of z-ideals in semirings and explores their properties, particularly focusing on their relationship with closure operations and distinguished ideal types like z-prime, z-semiprime, z-irreducible, and z-strongly irreducible ideals.

### Twisted Jacquet Modules of Sub-Quotients of Principal Series Representations of GL2(D) and a Conjecture of D. Prasad

This paper investigates the structure of twisted Jacquet modules of certain representations of GL2(D), where D is a division algebra over a non-archimedean local field, proving a conjecture by D. Prasad in the case of the quaternionic division algebra and providing dimension formulas and explicit descriptions in the depth-zero case.

### Unitary Branching Rules for the General Linear Lie Superalgebra (glm|n)

This paper establishes branching rules for finite-dimensional unitary simple modules of the general linear Lie superalgebra (glm|n), providing necessary and sufficient conditions for their decomposition into simple unitary glm|n-1-modules.

### The Equivalence of Koebe's Circle Domain Conjecture and a Weyl-Type Problem in Hyperbolic 3-Space

The Koebe circle domain conjecture, which posits that any planar domain is conformally equivalent to a circle domain, is equivalent to a specific formulation of the Weyl problem in hyperbolic 3-space, asserting that every complete hyperbolic surface of genus zero is isometric to the boundary of the hyperbolic convex hull of a circle type closed set in the sphere at infinity.