Tootfinder

This https://arxiv.org/abs/2412.12921 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Equivariant and Invariant Parametrized Topological Complexity
For a $G$-equivariant fibration $p \colon E\to B$, we introduce and study the invariant analogue of Cohen, Farber and Weinberger's parametrized topological complexity, called the invariant parametrized topological complexity. This notion generalizes the invariant topological complexity introduced by Lubawski and Marzantowicz. We establish the equivariant fibrewise homotopy invariance of this notion and derive several bounds, including a cohomological lower bound and a dimensional upper bound. A…

Morphisms and comorphisms of sites II - Distributors of sites
Axel Osmond, Olivia Caramello
https://arxiv.org/abs/2507.20932 https://arxiv.org/pdf/2507.209…

Morphisms and comorphisms of sites II - Distributors of sites
We introduce a notion of distributor of sites, involving suited analogs of flatness and cover-preservation, and show that this notion jointly generalizes those of morphism and comorphism of sites. Given two sites, we exhibit an adjunction between the category of distributors of sites between them and the category of geometric morphisms between the associated sheaf topoi; this adjunction restricts to an equivalence between geometric morphisms and continuous distributors of sites. We finally disc…

Evasive Random Walks and the Clairvoyant Demon
Aaron Abrams, Henry Landau, Zeph Landau, James Pommersheim, Eric Zaslow
https://arxiv.org/abs/2506.21929 htt…

Evasive Random Walks and the Clairvoyant Demon
A pair of random walks $(R,S)$ on the vertices of a graph $G$ is {\it successful} if two tokens can be scheduled (moving only one token at a time) to travel along $R$ and $S$ without colliding. We consider questions related to P. Winkler's {\it clairvoyant demon problem}, which asks whether for random walks $R$ and $S$ on $G$, $Pr[\ (R,S) \mbox{ is successful }] >0$. We introduce the notion of an {\it evasive} walk on $G$: a walk $S$ so that for a random walk $R$ on $G$, $Pr[\ (R,S) \mbox{ is s…

Defining ethically sourced code generation
Zhuolin Xu, Chenglin Li, Qiushi Li, Shin Hwei Tan
https://arxiv.org/abs/2507.19743 https://arxiv.org/pdf/2507.19…

Defining ethically sourced code generation
Several code generation models have been proposed to help reduce time and effort in solving software-related tasks. To ensure responsible AI, there are growing interests over various ethical issues (e.g., unclear licensing, privacy, fairness, and environment impact). These studies have the overarching goal of ensuring ethically sourced generation, which has gained growing attentions in speech synthesis and image generation. In this paper, we introduce the novel notion of Ethically Sourced Code …

Relating insplittings of 2-graphs and of textile systems
Samantha Brooker, Priyanga Ganesan, Elizabeth Gillaspy, Ying-Fen Lin, David Pask, Julia Plavnik
https://arxiv.org/abs/2506.21708

Relating insplittings of 2-graphs and of textile systems
The graphical operation of insplitting is key to understanding conjugacy of shifts of finite type (SFTs) in both one and two dimensions. In this paper, we consider two approaches to studying 2-dimensional SFTs: textile systems and rank-2 graphs. Nasu's textile systems describe all two-sided 2D SFTs up to conjugacy, whereas the 2-graphs (higher-rank graphs of rank 2) introduced by Kumjian and Pask yield associated C*-algebras. Both models have a naturally-associated notion of insplitting. We sho…

Fully Dynamic Spectral and Cut Sparsifiers for Directed Graphs
Yibin Zhao
https://arxiv.org/abs/2507.19632 https://arxiv.org/pdf/2507.19632

Fully Dynamic Spectral and Cut Sparsifiers for Directed Graphs
Recent years have seen extensive research on directed graph sparsification. In this work, we initiate the study of fast fully dynamic spectral and cut sparsification algorithms for directed graphs. We introduce a new notion of spectral sparsification called degree-balance preserving spectral approximation, which maintains the difference between the in-degree and out-degree of each vertex. The approximation error is measured with respect to the corresponding undirected Laplacian. This notion i…

Realization, Interpolation, Toeplitz corona and Extension Theorem for Tetrablock
Shubham Jain, Surjit Kumar, Milan Kumar Mal, Paramita Pramanick
https://arxiv.org/abs/2505.23492

Realization, Interpolation, Toeplitz corona and Extension Theorem for Tetrablock
We introduce a notion of Schur Agler class associated with the tetrablock and establish a realization theorem for this class. Furthermore, we provide a tetrablock analog of the interpolation theorem, the Toeplitz corona theorem, and an extension theorem.

Non-Hamiltonian 2-regular Digraphs
Munagala V. S. Ramanath
https://arxiv.org/abs/2507.21381 https://arxiv.org/pdf/2507.21381…

Non-Hamiltonian 2-regular Digraphs
In earlier papers, we showed a decomposition of 2-diregular digraphs (2-dds) and used it to provide some sufficient conditions for these graphs to be non-Hamiltonian; we also showed a close connection between the permanent and determinant of the adjacency matrices of these digraphs and gave some enumeration and generation results. In the present paper we extend the discussion to a larger class of digraphs, introduce the notions of routes and quotients and use them to provide additional criteria…

2-Segal sets and pseudomonoids in the bicategory of spans
Sophia E Marx, Rajan Amit Mehta
https://arxiv.org/abs/2505.22832 https://ar…

2-Segal sets and pseudomonoids in the bicategory of spans
In this survey article, we give an introduction to the notion of a 2-Segal set and prove that 2-Segal sets are equivalent to pseudomonoids in the bicategory of spans. The proof utilizes graphical techniques for 2-Segal sets and spans that should be useful in more general settings. There are procedures for obtaining an associative algebra from a 2-Segal set (satisfying finiteness conditions). We describe these procedures and give several examples of algebras arising from 2-Segal sets. Wherev…

On Weary Drivers, Records of Trees, and Parking Functions
Adri\'an Lillo, Mercedes Rosas, Stefan Trandafir
https://arxiv.org/abs/2506.22145 https://

On Weary Drivers, Records of Trees, and Parking Functions
This work builds on a decomposition of a Cayley tree using the notion of a record, a concept closely related to the blob encoding of a tree introduced by Kreweras and Moszkowski, and Picciotto, that we explore and extend. We provide an alternative definition of parking functions and derive from it a record-preserving bijection between Cayley trees and parking functions. Finally, we use this bijection to establish an equidistribution between a sextuple of statistics on Cayley trees and a corresp…