Tootfinder

Pullbacks and intersections in categories of graphs of groups
Jordi Delgado, Marco Linton, Jone Lopez de Gamiz Zearra, Mallika Roy, Pascal Weil
https://arxiv.org/abs/2508.04362 …

Pullbacks and intersections in categories of graphs of groups
We develop a categorical framework for studying graphs of groups and their morphisms, with emphasis on pullbacks. More precisely, building on classical work by Serre and Bass, we give an explicit construction of the so-called $\mathbb{A}$-product of two morphisms into a graph of groups $\mathbb{A}$ -- a graph of groups which, within the appropriate categorical setting, captures the intersection of subgroups of the fundamental group of $\mathbb{A}$. We show that, in the category of pointed graph…

Intertwining Algebras and Affine Hecke Algebras for Finite Central Extensions of Classical $p$-adic Groups with Application to Metaplectic Groups
Volker Heiermann, Chenyan Wu
https://arxiv.org/abs/2508.04303

Intertwining Algebras and Affine Hecke Algebras for Finite Central Extensions of Classical $p$-adic Groups with Application to Metaplectic Groups
For a finite central extension $\tilde{G}$ of a classical $p$-adic reductive group, we consider the endomorphism algebra of some induced projective generator à la Bernstein of the category of smooth representations of $\tilde{G}$. In the case where the Levi subgroups decompose, we can compute this algebra to get a result similar to the one previously obtained by the first author for classical $p$-adic groups, showing that this intertwining algebra is a twisted semi-direct product of an affine …

Generative AI for Object-Oriented Programming: Writing the Right Code and Reasoning the Right Logic
Gang Xu, Airong Wang, Yushan Pan
https://arxiv.org/abs/2508.05005 https://

Generative AI for Object-Oriented Programming: Writing the Right Code and Reasoning the Right Logic
We find ourselves in the midst of an explosion in artificial intelligence research, particularly with large language models (LLMs). These models have diverse applications spanning finance, commonsense knowledge graphs, medicine, and visual analysis. In the world of Object-Oriented Programming(OOP), a robust body of knowledge and methods has been developed for managing complex tasks through object-oriented thinking. However, the intersection of LLMs with OOP remains an underexplored territory. E…

On the Isomorphism Relation for Omnigenous Locally Finite Groups
Su Gao, Feng Li
https://arxiv.org/abs/2507.03907 https://arxiv.org/p…

On the Isomorphism Relation for Omnigenous Locally Finite Groups
The concept of an omnigenous locally finite group was introduced in [2] as a generalization of Hall's universal countable locally finite group. In this paper we show that the class of all countable omnigenous locally finite groups is Borel complete, hence it has the maximum Borel cardinality of isomorphism types among all countable structures. [2] M. Etedadialiabadi, S. Gao, F. Le Maître, J. Melleray, Dense locally finite subgroups of automorphism groups of ultraextensive spaces, Adv. Math. …

Improved lower bounds on the maximum size of graphs with girth 5
Jan Goedgebeur, Jorik Jooken, Gwena\"el Joret, Tibo Van den Eede
https://arxiv.org/abs/2508.05562 https://

Improved lower bounds on the maximum size of graphs with girth 5
We present a new algorithm for improving lower bounds on $ex(n;\{C_3,C_4\})$, the maximum size (number of edges) of an $n$-vertex graph of girth at least 5. The core of our algorithm is a variant of a hill-climbing heuristic introduced by Exoo, McKay, Myrvold and Nadon (2011) to find small cages. Our algorithm considers a range of values of $n$ in multiple passes. In each pass, the hill-climbing heuristic for a specific value of $n$ is initialized with a few graphs obtained by modifying near-ex…

GASP: A Gradient-Aware Shortest Path Algorithm for Boundary-Confined Visualization of 2-Manifold Reeb Graphs
Sefat Rahman, Tushar M. Athawale, Paul Rosen
https://arxiv.org/abs/2508.05524

GASP: A Gradient-Aware Shortest Path Algorithm for Boundary-Confined Visualization of 2-Manifold Reeb Graphs
Reeb graphs are an important tool for abstracting and representing the topological structure of a function defined on a manifold. We have identified three properties for faithfully representing Reeb graphs in a visualization. Namely, they should be constrained to the boundary, compact, and aligned with the function gradient. Existing algorithms for drawing Reeb graphs are agnostic to or violate these properties. In this paper, we introduce an algorithm to generate Reeb graph visualizations, cal…

Online Sparsification of Bipartite-Like Clusters in Graphs
Joyentanuj Das, Suranjan De, He Sun
https://arxiv.org/abs/2508.05437 https://arxiv.org/pdf/2508.…

Online Sparsification of Bipartite-Like Clusters in Graphs
Graph clustering is an important algorithmic technique for analysing massive graphs, and has been widely applied in many research fields of data science. While the objective of most graph clustering algorithms is to find a vertex set of low conductance, a sequence of recent studies highlights the importance of the inter-connection between vertex sets when analysing real-world datasets. Following this line of research, in this work we study bipartite-like clusters and present efficient and onlin…

SSEmb: A Joint Structural and Semantic Embedding Framework for Mathematical Formula Retrieval
Ruyin Li, Xiaoyu Chen
https://arxiv.org/abs/2508.04162 https://

SSEmb: A Joint Structural and Semantic Embedding Framework for Mathematical Formula Retrieval
Formula retrieval is an important topic in Mathematical Information Retrieval. We propose SSEmb, a novel embedding framework capable of capturing both structural and semantic features of mathematical formulas. Structurally, we employ Graph Contrastive Learning to encode formulas represented as Operator Graphs. To enhance structural diversity while preserving mathematical validity of these formula graphs, we introduce a novel graph data augmentation approach through a substitution strategy. Sema…

The Supermarket Model on a Dynamic Regular Hypergraph
John Fernley, Bal\'azs Gerencs\'er
https://arxiv.org/abs/2507.03931 https://

The Supermarket Model on a Dynamic Regular Hypergraph
The supermarket model is a system of $n$ queues each with serving rates $1$ and arrival rates $λ$ per vertex, where tasks will move on arrival to the shortest adjacent queue. We consider the supermarket model in the small $λ$ regime on a large dynamic configuration hypergraph with stubs swapping their hyperedge membership at rate $κ$. This interpolates previous investigations of the supermarket model on static graphs of bounded degree (where an exponential tail produces a logarithmic queue…