Tootfinder

106 for a fitness fix! Way to redline 69
A new p.r. today in set one: 106 reps at the current level (35-cm offset at the feet; parallettes for full range of motion). The middle section here is sped up to 9x speed for your entertainment and viewing ease. I will do a second set in the evening. My goal is to hit two sets of 100 or more in a day. I hope to hit that by my 70th birthday in nine weeks.

Active Learning of Symbolic Mealy Automata
Kengo Irie, Masaki Waga, Kohei Suenaga
https://arxiv.org/abs/2509.14694 https://arxiv.org/pdf/2509.14694

Active Learning of Symbolic Mealy Automata
We propose $Λ^*_M$-an active learning algorithm that learns symbolic Mealy automata, which support infinite input alphabets and multiple output characters. Each of these two features has been addressed separately in prior work. Combining these two features poses a challenge in learning the outputs corresponding to potentially infinite sets of input characters at each state. To address this challenge, we introduce the notion of essential input characters, a finite set of input characters that i…

Generalised Reachability Games Revisited
Sougata Bose (UMONS - Universit\'e de Mons), Daniel Hausmann (University of Liverpool), Soumyajit Paul (University of Liverpool), Sven Schewe (University of Liverpool), Tansholpan Zhanabekova (University of Liverpool)
https://arxiv.org/abs/2509.14091

Generalised Reachability Games Revisited
Classic reachability games on graphs are zero-sum games, where the goal of one player, Eve, is to visit a vertex from a given target set, and that of other player, Adam, is to prevent this. Generalised reachability games, studied by Fijalkow and Horn, are a generalisation of reachability objectives, where instead of a single target set, there is a family of target sets and Eve must visit all of them in any order. In this work, we further study the complexity of solving two-player games on graph…

An Adaptive Sampling Algorithm for Level-set Approximation
Matteo Croci, Abdul-Lateef Haji-Ali, Ian C. J. Powell
https://arxiv.org/abs/2509.14896 https://a…

An Adaptive Sampling Algorithm for Level-set Approximation
We propose a new numerical scheme for approximating level-sets of Lipschitz multivariate functions which is robust to stochastic noise. The algorithm's main feature is an adaptive grid-based stochastic approximation strategy which automatically refines the approximation over regions close to the level set. This strategy combines a local function approximation method with a noise reduction scheme and produces $\varepsilon$-accurate approximations with an expected cost complexity reduction of $\v…

Significant inference and confidence sets for graphical models
P. A. Koldanov, A. P. Koldanov
https://arxiv.org/abs/2509.12292 https://arxiv.org/pdf/2509.1…

Significant inference and confidence sets for graphical models
The problem of identifying statistically significant inferences about the structure of the graphical model is considered, along with the related task of constructing a confidence set for a graphical model. It has been proven that the procedure for constructing such set is equivalent to the procedure for simultaneous testing of hypotheses and alternatives regarding the composition of the graphical model. Some variants of the simultaneous testing of hypotheses and alternatives are discussed. It i…

An averaging result for union-closed families of sets
Christopher Bouchard
https://arxiv.org/abs/2509.12537 https://arxiv.org/pdf/2509.12537

An averaging result for union-closed families of sets
Let $\mathcal{A}$ be a union-closed family of sets with base set $b(\mathcal{A})=\bigcup_{A \in \mathcal{A}}A$ denoted by $[n]=\{1, \cdots, n\}$, and for any real $x>0$, let $\mathcal{A}_{

Dual Shellability of Admissible Set and Cohen-Macaulayness of Local Models
Xuhua He, Qingchao Yu
https://arxiv.org/abs/2509.11581 https://arxiv.org/pdf/250…

Dual Shellability of Admissible Set and Cohen-Macaulayness of Local Models
We prove Görtz's combinatorial conjecture \cite{Go01} on dual shellability of admissible sets in Iwahori-Weyl groups, proving that the augmented admissible set $\widehat{\mathrm{Adm}}(μ)$ is dual shellable for any dominant coweight $μ$. This provides a uniform, elementary approach to establishing Cohen-Macaulayness of the special fibers of the local models with Iwahori level structure for all reductive groups-including residue characteristic $2$ and non-reduced root systems-circumventing geo…

sp_colocation: Social co-locations (2018)
Network of colocations between peoople, based on the information on which RFID readers received information from the RFID tags. Namely, we define two individuals to be in co-presence if the same exact set of readers have received signals from both individuals during a 20s time window.
This network has 332 nodes and 18613039 edges.
Tags: Social, Offline, Unweighted, Weighted, Temporal, Metadata

Dynamical Perturbing and $C^*$-algebra Lifting Problems
Samantha Pilgrim
https://arxiv.org/abs/2509.13599 https://arxiv.org/pdf/2509.13599

Dynamical Perturbing and $C^*$-algebra Lifting Problems
We initiate a study of a new kind of topological-dynamical stability by investigating when permutations of finite sets within a metric space which are close to being actions by a group can be perturbed to genuine actions. We show these perturbing problems can have solutions in the case of dynamics on the Cantor set, and develop a theoretical framework for potentially producing examples of other perturbing problems with no solution. We also investigate connections with semiprojectivity of $C^*$-…

G-CSEA: A Graph-Based Conflict Set Extraction Algorithm for Identifying Infeasibility in Pseudo-Boolean Models
Kanishk Garg, Saranya D., Sanal Kumar, Saurabh Singh, Anupam Purwar
https://arxiv.org/abs/2509.13203

G-CSEA: A Graph-Based Conflict Set Extraction Algorithm for Identifying Infeasibility in Pseudo-Boolean Models
Workforce scheduling involves a variety of rule-based constraints-such as shift limits, staffing policies, working hour restrictions, and many similar scheduling rules-which can interact in conflicting ways, leading to infeasible models. Identifying the underlying causes of such infeasibility is critical for resolving scheduling issues and restoring feasibility. A common diagnostic approach is to compute Irreducible Infeasible Subsets (IISs): minimal sets of constraints that are jointly infeasi…

CS-FLEURS: A Massively Multilingual and Code-Switched Speech Dataset
Brian Yan, Injy Hamed, Shuichiro Shimizu, Vasista Lodagala, William Chen, Olga Iakovenko, Bashar Talafha, Amir Hussein, Alexander Polok, Kalvin Chang, Dominik Klement, Sara Althubaiti, Puyuan Peng, Matthew Wiesner, Thamar Solorio, Ahmed Ali, Sanjeev Khudanpur, Shinji Watanabe, Chih-Chen Chen, Zhen Wu, Karim Benharrak, Anuj Diwan, Samuele Cornell, Eunjung Yeo, Kwanghee Choi, Carlos Carvalho, Karen Rosero

CS-FLEURS: A Massively Multilingual and Code-Switched Speech Dataset
We present CS-FLEURS, a new dataset for developing and evaluating code-switched speech recognition and translation systems beyond high-resourced languages. CS-FLEURS consists of 4 test sets which cover in total 113 unique code-switched language pairs across 52 languages: 1) a 14 X-English language pair set with real voices reading synthetically generated code-switched sentences, 2) a 16 X-English language pair set with generative text-to-speech 3) a 60 {Arabic, Mandarin, Hindi, Spanish}-X langu…

Replaced article(s) found for math.HO. https://arxiv.org/list/math.HO/new
[1/1]:
- Descriptions of Cantor Sets: A Set-Theoretic Survey and Open Problems
Mohsen Soltanifar

Series B, Episode 11 - Gambit
SERVALAN: He'll soon come round. Travis is very strong. [She goes to inspect Travis.]
JARRIERE: Oh, he once worked for you?
SERVALAN: Yes.
JARRIERE: And now he's an outlaw? Uh, what happened?
https://blake.torpidity.net/m/211/202 B7B2…

Optimal Conformal Prediction, E-values, Fuzzy Prediction Sets and Subsequent Decisions
Nick W. Koning, Sam van Meer
https://arxiv.org/abs/2509.13130 https://

Optimal Conformal Prediction, E-values, Fuzzy Prediction Sets and Subsequent Decisions
We make three contributions to conformal prediction. First, we propose fuzzy conformal confidence sets that offer a degree of exclusion, generalizing beyond the binary inclusion/exclusion offered by classical confidence sets. We connect fuzzy confidence sets to e-values to show this degree of exclusion is equivalent to an exclusion at different confidence levels, capturing precisely what e-values bring to conformal prediction. We show that a fuzzy confidence set is a predictive distribution wit…

Large cardinals beyond HOD
Juan Pablo Aguilera, Joan Bagaria, Gabriel Goldberg, Philipp L\"ucke
https://arxiv.org/abs/2509.10254 https://arxiv.org/pdf…

Large cardinals beyond HOD
Exacting and ultraexacting cardinals are large cardinal numbers compatible with the Zermelo-Fraenkel axioms of set theory, including the Axiom of Choice. In contrast with standard large cardinal notions, their existence implies that the set-theoretic universe V is not equal to Gödel's subuniverse of Hereditarily Ordinal Definable (HOD) sets. We prove that the existence of an ultraexacting cardinal is equiconsistent with the well-known axiom I0; moreover, the existence of ultraexacting cardinal…

Loss-aware distributionally robust optimization via trainable optimal transport ambiguity sets
Jonas Ohnemus, Marta Fochesato, Riccardo Zuliani, John Lygeros
https://arxiv.org/abs/2509.12689

Loss-aware distributionally robust optimization via trainable optimal transport ambiguity sets
Optimal-Transport Distributionally Robust Optimization (OT-DRO) robustifies data-driven decision-making under uncertainty by capturing the sampling-induced statistical error via optimal transport ambiguity sets. The standard OT-DRO pipeline consists of a two-step procedure, where the ambiguity set is first designed and subsequently embedded into the downstream OT-DRO problem. However, this separation between uncertainty quantification and optimization might result in excessive conservatism. We …

Hilbert spaces admit no finitary discrete imaginaries
Ruiyuan Chen, Isabel Trindade
https://arxiv.org/abs/2509.11321 https://arxiv.org/pdf/2509.11321

Hilbert spaces admit no finitary discrete imaginaries
We prove that every functor from the category of Hilbert spaces and linear isometric embeddings to the category of sets which preserves directed colimits must be essentially constant on all infinite-dimensional spaces. In other words, every finitary set-valued imaginary over the theory of Hilbert spaces, in a broad signature-independent sense, must be essentially trivial. This extends a result and answers a question by Lieberman--Rosický--Vasey, who showed that no such functor on the supercate…

Reconstructing Finite Ordered Sets from Higher Exponentials
George Gr\"atzer
https://arxiv.org/abs/2509.12659 https://arxiv.org/pdf/2509.12659

Reconstructing Finite Ordered Sets from Higher Exponentials
For the finite ordered sets $A, D$, write $A^D$ for the ordered set of isotone maps $D \to A$ with the pointwise order. It was proved in earlier work that the order structure of $A^A$ determines~$A$ up to isomorphism. In this note we extend the result to higher function ordered sets such as $A^{(A^A)}$ and $(A^A)^A$. Our main theorem shows that the structure of $A^D$ determines~$A$.

Polynomial convexity of $\bar\partial$-flat perturbations of totally real sets
Leandro Arosio, H{\aa}kan Samuelsson Kalm, Erlend F. Wold
https://arxiv.org/abs/2510.11943 https:/…

Polynomial convexity of $\bar\partial$-flat perturbations of totally real sets
We show that if $X$ is a totally real $d$-dimensional manifold attached to a polynomially convex compact set $K$ in $\mathbb{C}^n$, $d

Hausdorff measure of cartesian product of Cantor sets
Siyuan Guo, Taylor Jones
https://arxiv.org/abs/2510.09701 https://arxiv.org/pdf/2510.09701

Hausdorff measure of cartesian product of Cantor sets
Hausdorff measure and Hausdorff dimension are useful tools to describe fractals. This paper investigates the bounds on the $d\log_32$-dimensional Hausdorff measure of the $d$-fold Cartesian product of the $1/3$ Cantor set, $\mathcal C^d$. By applying known theorems on the Hausdorff measure of fractals satisfying the strong open set condition and generalizing what has been done on $\mathcal C^2$, we compute stricter upper and lower bounds for the Hausdorff measure of $\mathcal C^d$ for several s…

Parameterized Algorithms for Diversity of Networks with Ecological Dependencies
Mark Jones, Jannik Schestag
https://arxiv.org/abs/2510.09512 https://arxiv.…

Parameterized Algorithms for Diversity of Networks with Ecological Dependencies
For a phylogenetic tree, the phylogenetic diversity of a set A of taxa is the total weight of edges on paths to A. Finding small sets of maximal diversity is crucial for conservation planning, as it indicates where limited resources can be invested most efficiently. In recent years, efficient algorithms have been developed to find sets of taxa that maximize phylogenetic diversity either in a phylogenetic network or in a phylogenetic tree subject to ecological constraints, such as a food web. Ho…

Restarting #Running, take 37 🎬
Busyness and niggling knee pain derailed me from my last efforts. Then inertia set in and everything becomes harder.
But, here we start anew. One foot in front of the other.

Reciprocal relations for orthogonal quantum matrices
Pavel Pyatov, Oleg Ogievetsky
https://arxiv.org/abs/2510.09811 https://arxiv.org/pdf/2510.09811…

Reciprocal relations for orthogonal quantum matrices
For the family of the orthogonal quantum matrix algebras we investigate the structure of their characteristic subalgebras -- special commutative subalgebras, which for the subfamily of the reflection equation algebras appear to be central. In [OP1] we described three generating sets of the characteristic subalgebras of the symplectic and orthogonal quantum matrix algebras. One of these -- the set of the elementary sums -- is finite. In the symplectic case the elementary sums are in general alge…

A PCA-based Data Prediction Method
Peteris Daugulis, Vija Vagale, Emiliano Mancini, Filippo Castiglione
https://arxiv.org/abs/2510.09246 https://arxiv.org/…

A PCA-based Data Prediction Method
The problem of choosing appropriate values for missing data is often encountered in the data science. We describe a novel method containing both traditional mathematics and machine learning elements for prediction (imputation) of missing data. This method is based on the notion of distance between shifted linear subspaces representing the existing data and candidate sets. The existing data set is represented by the subspace spanned by its first principal components. Solutions for the case of th…

A new description of uniformly spread discrete sets
A. Dudko, S. Favorov
https://arxiv.org/abs/2510.11061 https://arxiv.org/pdf/2510.11061

A new description of uniformly spread discrete sets
We prove that each discrete set in the Euclidean space that has bounded changes under every translation is a bounded perturbation of a square lattice, i.e., a uniformly spread set in the sense of Laszkovich. In particular, the support of every Fourier quasicrystal with unit masses is uniformly spread.

The isoperimetric inequality for the capillary energy outside convex sets
N. Fusco, V. Julin, M. Morini, A. Pratelli
https://arxiv.org/abs/2509.10200 https://

The isoperimetric inequality for the capillary energy outside convex sets
We study the isoperimetric problem for capillary hypersurfaces with a general contact angle $θ\in (0, π)$, outside arbitrary convex sets. We prove that the capillary energy of any surface supported on any such convex set is larger than that of a spherical cap with the same volume and the same contact angle on a flat support, and we characterize the equality cases. This provides a complete solution to the isoperimetric problem for capillary surfaces outside convex sets at arbitrary contact ang…

Concentrated sets and the Hurewicz property
Valentin Haberl, Piotr Szewczak, Lyubomyr Zdomskyy
https://arxiv.org/abs/2511.09320 https://arxiv.org/pdf/2511.09320 https://arxiv.org/html/2511.09320
arXiv:2511.09320v1 Announce Type: new
Abstract: A set of reals $X$ is $\mathfrak{b}$-concentrated if it has cardinality at least $\mathfrak{b}$ and it contains a countable set $D\subseteq X$ such that each closed subset of $X$ disjoint with $D$ has size smaller than $\mathfrak{b}$. We present ZFC results about structures of $\mathfrak{b}$-concentrated sets with the Hurewicz covering property using semifilters. Then we show that assuming that the semifilter trichotomy holds, then each $\mathfrak{b}$-concentrated set is Hurewicz and even productively Hurewicz. We also show that the appearance of Hurewicz $\mathfrak{b}$-concentrated sets under the semifilter trichotomy is somewhat specific and the situation in the Laver model for the consitency of the Borel Conjecture is different.
toXiv_bot_toot

Constructions of Sequences of Alternating Sum and Difference Dominated Sets
Yorick Herrmann, Connor Hill, Merlin Phillips, Daniel Flores, Steven J. Miller, Steven Senger
https://arxiv.org/abs/2509.00792

Constructions of Sequences of Alternating Sum and Difference Dominated Sets
A More Sums Than Difference (MSTD) set is a finite set of integers $A$ where the cardinality of its sumset, $A+A$, is greater than the cardinality of its difference set, $A-A$. Since addition is commutative while subtraction isn't, it was conjectured that MSTD sets are rare. As Martin and O'Bryant proved a small (but positive) percentage are MSTD, it is natural to ask what additional properties can we impose on a chain of MSTD sets; in particular, can we construct a sequence of sets alternating…

Similarity between the Multibrot set and the Julia set of correspondences at Misiurewicz points
Carlos Siqueira
https://arxiv.org/abs/2509.07266 https://ar…

Similarity between the Multibrot set and the Julia set of correspondences at Misiurewicz points
We study the fine structure of the parameter space of the unicritical family of algebraic correspondences $z^r + c$, where $r > 1$ is a rational exponent. Building on Tan Lei's result regarding the similarity between the Mandelbrot set and Julia sets in the quadratic family, we prove that the Julia set of the correspondence is asymptotically self-similar about every Misiurewicz point. Assuming that the transversality condition holds at a Misiurewicz parameter $a \in \mathbb{C}$, we prove that…

Data-Model Co-Evolution: Growing Test Sets to Refine LLM Behavior
Minjae Lee, Minsuk Kahng
https://arxiv.org/abs/2510.12728 https://arxiv.org/pdf/2510.1272…

Data-Model Co-Evolution: Growing Test Sets to Refine LLM Behavior
A long-standing challenge in machine learning has been the rigid separation between data work and model refinement, enforced by slow fine-tuning cycles. The rise of Large Language Models (LLMs) overcomes this historical barrier, allowing applications developers to instantly govern model behavior by editing prompt instructions. This shift enables a new paradigm: data-model co-evolution, where a living test set and a model's instructions evolve in tandem. We operationalize this paradigm in an int…

Three-Neighbour Bootstrap Percolation in Thin Three-Dimensional Grids
Will Dolphin, Peter J. Dukes
https://arxiv.org/abs/2509.13589 https://arxiv.org/pdf/2…

Three-Neighbour Bootstrap Percolation in Thin Three-Dimensional Grids
We improve the status of the problem of determining minimum-sized percolating sets in $a \times b \times c$ grids under the $3$-neighbour process. Using several new constructions, we show that optimal percolating sets exist whenever $\min(a,b,c) \ge 7$. As an important step toward this, we also show that all grids with $\min(a,b,c) \ge 4$ have a percolating set whose size exactly achieves the lower bound $(ab+ac+bc)/3$ whenever this value is an integer.

Identifying Imperfect Clones in Elections
Piotr Faliszewski, Lukasz Janeczko, Grzegorz Lisowski, Kristyna Pekarkova, Ildiko Schlotter
https://arxiv.org/abs/2509.11261 https://…

Identifying Imperfect Clones in Elections
A perfect clone in an ordinal election (i.e., an election where the voters rank the candidates in a strict linear order) is a set of candidates that each voter ranks consecutively. We consider different relaxations of this notion: independent or subelection clones are sets of candidates that only some of the voters recognize as a perfect clone, whereas approximate clones are sets of candidates such that every voter ranks their members close to each other, but not necessarily consecutively. We e…

Geometry of dyadic polygons I: the structure of dyadic triangles
A. Mu\'cka, A. Romanowska
https://arxiv.org/abs/2510.05467 https://arxiv.org/pdf/2510.…

Geometry of dyadic polygons I: the structure of dyadic triangles
Dyadic rationals are rationals whose denominator is a power of 2. A dyadic n-dimensional convex set is defined as the intersection with n-dimensional dyadic space of an n-dimensional real convex set. Such a dyadic convex set is said to be a dyadic n-dimensional polytope if the real convex set is a polytope whose vertices lie in the dyadic space. Dyadic convex sets are described as subalgebras of reducts of faithful affine spaces over the ring of dyadic numbers, or equivalently as commutative, e…

Smooth Uncertainty Sets: Dependence of Uncertain Parameters via a Simple Polyhedral Set
Noam Goldberg, Michael Poss, Shimrit Shtern
https://arxiv.org/abs/2510.08843 https://

Smooth Uncertainty Sets: Dependence of Uncertain Parameters via a Simple Polyhedral Set
We propose a novel polyhedral uncertainty set for robust optimization, termed the smooth uncertainty set, which captures dependencies of uncertain parameters by constraining their pairwise differences. The bounds on these differences may be dictated by the underlying physics of the problem and may be expressed by domain experts. When correlations are available, the bounds can be set to ensure that the associated probabilistic constraints are satisfied for any given probability. We explore spe…

The sun's out again in Ōtautahi! The local skink community is taking full advantage!
#biodiversity #nativeSkinks #lizardarium

Weekend Reads
* Monitoring AS-SETs
https://blog.cloudflare.com/monitoring-as-sets-and-why-they-matter/
* IX LAN broadcast traffic

Monitoring AS-SETs and why they matter
AS-SETs, for better or worse, are critical data sources that help build BGP route filters across the Internet. Because these AS-SET objects are based solely on trust, they can be misused. We will cover some of the reasons why operators need to monitor the AS-SET memberships for their ASN, and now Cloudflare Radar can help.

Geno Smith’s performance vs. Patriots sets new Raiders record https://bolavip.com/en/nfl/geno-smiths-performance-vs-patriots-sets-new-raiders-record

Monoid Structures on Indexed Containers
Michele De Pascalis, Tarmo Uustalu, Niccol\`o Veltr\`i
https://arxiv.org/abs/2509.25879 https://arxiv.org/pdf/2509.…

Monoid Structures on Indexed Containers
Containers represent a wide class of type constructions relevant for functional programming and (co)inductive reasoning. Indexed containers generalize this notion to better fit the scope of dependently typed programming. When interpreting types to be sets, a container describes an endofunctor on the category of sets while an I-indexed container describes an endofunctor on the category Set^I of I-indexed families of sets. We consider the monoidal structure on the category of I-indexed containe…

AuraSight: Generating Realistic Social Media Data
Lynnette Hui Xian Ng, Bianca N. Y. Kang, Kathleen M. Carley
https://arxiv.org/abs/2509.08927 https://arxi…

AuraSight: Generating Realistic Social Media Data
This document details the narrative and technical design behind the process of generating a quasi-realistic set X data for a fictional multi-day pop culture episode (AuraSight). Social media post simulation is essential towards creating realistic training scenarios for understanding emergent network behavior that formed from known sets of agents. Our social media post generation pipeline uses the AESOP-SynSM engine, which employs a hybrid approach of agent-based and generative artificial intell…

FA-modules of holomorphic forms on $\overline{\mathcal{M}}_{g,n}$
Samir Canning, Hannah Larson, Sam Payne, Thomas Willwacher
https://arxiv.org/abs/2509.08774 https://

FA-modules of holomorphic forms on $\overline{\mathcal{M}}_{g,n}$
For fixed genus g and varying finite marking set A, the gluing and forgetful maps give the spaces of holomorphic forms on the moduli space of stable A-marked curves of genus g has the structure of an FA-module, i.e., a functor from the category of finite sets to vector spaces. We prove that the resulting FA-modules of holomorphic k-forms are simple, for k less than or equal to 18, whenever they are nonzero. Conditional upon the conjectured vanishing of holomorphic 19-forms and 20-forms in genus…

Gene-R1: Reasoning with Data-Augmented Lightweight LLMs for Gene Set Analysis
Zhizheng Wang, Yifan Yang, Qiao Jin, Zhiyong Lu
https://arxiv.org/abs/2509.10575 https://

Gene-R1: Reasoning with Data-Augmented Lightweight LLMs for Gene Set Analysis
The gene set analysis (GSA) is a foundational approach for uncovering the molecular functions associated with a group of genes. Recently, LLM-powered methods have emerged to annotate gene sets with biological functions together with coherent explanatory insights. However, existing studies primarily focus on proprietary models, which have been shown to outperform their open-source counterparts despite concerns over cost and data privacy. Furthermore, no research has investigated the application …

Volume functions and boundary data of 3-dimensional hyperbolic manifolds
Jean-Marc Schlenker
https://arxiv.org/abs/2510.05627 https://arxiv.org/pdf/2510.05…

Volume functions and boundary data of 3-dimensional hyperbolic manifolds
We review recent progress on two closely related sets of questions concerning convex co-compact hyperbolic manifolds, or convex domains in those manifolds, such as their convex core. The first set of questions is to what extent the hyperbolic metric on such a manifold is uniquely determined by either of two possible geometric data on their boundary. The second aspect is the ``volume'' associated to such a manifold, such as the renormalized volume of a convex co-compact hyperbolic manifold. The …

$k$-Fair Coalitions in Graphs
Abbas Jafari, Saeid Alikhani
https://arxiv.org/abs/2509.11358 https://arxiv.org/pdf/2509.11358…

$k$-Fair Coalitions in Graphs
Let $G = (V,E)$ be a simple graph. A subset $S \subseteq V$ is called a $k$-fair dominating set if every vertex not in $S$ has exactly $k$ neighbors in $S$. Two disjoint sets $A, B \subseteq V$ form a $k$-fair coalition of $G$ if neither $A$ nor $B$ is a $k$-fair dominating set and the union $A \cup B$ is a $k$-fair dominating set of $G$. A partition $π= \{V_1, V_2, \ldots, V_m\}$ of $V$ is called a $k$-fair coalition partition, if every set $V_i\inπ$, either $V_i$ is a $k$-fair dominating se…

Non-Asymptotic Analysis of Efficiency in Conformalized Regression
Yunzhen Yao, Lie He, Michael Gastpar
https://arxiv.org/abs/2510.07093 https://arxiv.org/p…

Non-Asymptotic Analysis of Efficiency in Conformalized Regression
Conformal prediction provides prediction sets with coverage guarantees. The informativeness of conformal prediction depends on its efficiency, typically quantified by the expected size of the prediction set. Prior work on the efficiency of conformalized regression commonly treats the miscoverage level $α$ as a fixed constant. In this work, we establish non-asymptotic bounds on the deviation of the prediction set length from the oracle interval length for conformalized quantile and median regre…

Regularity of cylindrical singular sets of mean curvature flow
Ao Sun, Zhihan Wang, Jinxin Xue
https://arxiv.org/abs/2509.01707 https://arxiv.org/pdf/2509.…

Regularity of cylindrical singular sets of mean curvature flow
In this paper, we study the $k$-cylindrical singular set of mean curvature flow in $\mathbb R^{n+1}$ for each $1\leq k\leq n-1$. We prove that they are locally contained in a $k$-dimensional $C^{2,α}$-submanifold after removing some lower-dimensional parts. Moreover, if the $k$-cylindrical singular set is a $k$-submanifold, then its curvature is determined by the asymptotic profile of the flow at these singularities. As a byproduct, we provide a detailed asymptotic profile and graphical radius…

Elementary differentials from multi-indices to rooted trees
Yvain Bruned, Paul Laubie
https://arxiv.org/abs/2509.13118 https://arxiv.org/pdf/2509.13118

Elementary differentials from multi-indices to rooted trees
Rooted trees are essential for describing numerical schemes via the so-called B-series. They have also been used extensively in rough analysis for expanding solutions of singular Stochastic Partial Differential Equations (SPDEs). When one considers scalar-valued equations, the most efficient combinatorial set is multi-indices. In this paper, we investigate the existence of intermediate combinatorial sets that will lie between multi-indices and rooted trees. We provide a negative result stating …

Adaptive Coverage Policies in Conformal Prediction
Etienne Gauthier, Francis Bach, Michael I. Jordan
https://arxiv.org/abs/2510.04318 https://arxiv.org/pdf…

Adaptive Coverage Policies in Conformal Prediction
Traditional conformal prediction methods construct prediction sets such that the true label falls within the set with a user-specified coverage level. However, poorly chosen coverage levels can result in uninformative predictions, either producing overly conservative sets when the coverage level is too high, or empty sets when it is too low. Moreover, the fixed coverage level cannot adapt to the specific characteristics of each individual example, limiting the flexibility and efficiency of thes…

The quasi-Assouad dimension of $(1,2t)$-Furstenberg sets in $\mathbb{R}^3$ is extremized by sticky sets
Sam Craig
https://arxiv.org/abs/2510.06462 https://…

The quasi-Assouad dimension of $(1,2t)$-Furstenberg sets in $\mathbb{R}^3$ is extremized by sticky sets
A $(1,2t)$-Furstenberg set in $\mathbb{R}^3$ is naturally defined as a set containing a union of unit line segments forming a $2t$-dimensional subset of the affine Grassmannian in $\mathbb{R}^3$ and satisfying a suitable variant of the Frostman Convex Wolff Axiom. Some of these sets have a multi-scale self-similarity property called stickiness. We investigate the extremizers of the quasi-Assouad dimension of $(1,2t)$-Furstenberg sets, a slightly stronger variant of the Assouad dimension. We pro…

SU(N) integrals and tau functions
A. Yu. Orlov
https://arxiv.org/abs/2509.20213 https://arxiv.org/pdf/2509.20213 …

SU(N) integrals and tau functions
We present a family of solvable multi-matrix models associated with an arbitrary embedded graph $Γ$ with a single vertex. The graph with $n$ edges is equipped with $2n$ corner matrices. The partition function of each member of the family depends on the set of eigenvalues of monodromies of corner matrices around the vertices of the dual graph $Γ^*$ and sets of parameters attached to each vertex of $Γ$. We select the cases where the partition function of a model is a tau function of KP, 2KP an…

Factorization of power GCD matrices and power LCM matrices on certain gcd-closed sets
Guangyan Zhu, Yuanyuan Luo, Jixiang Wan
https://arxiv.org/abs/2510.05595 https://

Factorization of power GCD matrices and power LCM matrices on certain gcd-closed sets
For integers $x$ and $y$, $(x, y)$ and $[x, y]$ stand for the greatest common divisor and the least common multiple of $x$ and $y$ respectively. Denote by $|T|$ the number of elements of a finite set $T$. Let $a,b$ and $n$ be positive integers and let $S=\{x_1, \cdots, x_n\}$ be a set of $n$ distinct positive integers. We denote by $(S^a)$ (resp. $[S^a]$) the $n\times n$ matrix having the $a$th power of $(x_i,x_j)$ (resp. $[x_i,x_j]$) as its $(i,j)$-entry. For any $x\in S$, define $G_…

On roundness of rotation sets
Boris Perrot, Jan Boro\'nski, Alex Clark
https://arxiv.org/abs/2510.08235 https://arxiv.org/pdf/2510.08235

On roundness of rotation sets
Motivated by the question whether a round disk can be realized as the rotation set of a torus diffeomorphism, we study the roundness of rotation sets of a parametric family of torus diffeomorphisms $F_ρ$, where the parameter $ρ$ ranges over irrational numbers in $(0,1)$. Each $F_ρ$ is a Kwapisz-like diffeomorphism with a 2-dimensional non-polygonal rotation set $$Λ'_ρ= \operatorname{conv}\left(\left\{(\pm\frac{\lceil mρ\rceil}{m+n+1}, \pm\frac{\lceil nρ\rceil}{m+n+1}): m, n \in \mathbb…

Combined-distance-based score function of cognitive fuzzy sets and its application in lung cancer pain evaluation
Lisheng Jiang, Tianyu Zhang, Shiyu Yan, Ran Fang
https://arxiv.org/abs/2509.08239

Combined-distance-based score function of cognitive fuzzy sets and its application in lung cancer pain evaluation
In decision making, the cognitive fuzzy set (CFS) is a useful tool in expressing experts' complex assessments of alternatives. The distance of CFS, which plays an important role in decision analyses, is necessary when the CFS is applied in solving practical issues. However, as far as we know, the studies on the distance of CFS are few, and the current Minkowski distance of CFS ignores the hesitancy degree of CFS, which might cause errors. To fill the gap of the studies on the distance of CFS, b…

The maximum product of sizes of cross-\(t\)-intersecting families
Jingjun Bao, Lijun Ji
https://arxiv.org/abs/2510.11724 https://arxiv.org/pdf/2510.11724…

The maximum product of sizes of cross-\(t\)-intersecting families
Two families of sets \(\mathcal{A}\) and \(\mathcal{B}\) are called \emph{cross-\(t\)-intersecting} if \(|A \cap B| \geq t\) for all \(A \in \mathcal{A}\) and \(B \in \mathcal{B}\). Determining the maximum product of sizes for such cross-\(t\)-intersecting families is an active problem in extremal set theory. In this paper, we verify the following cross-\(t\)-intersecting version of the Erdős-Ko-Rado theorem: For \(k\geq l \geq t \geq 3\) and \(\min\{m,n\} \geq (t+1)(k-t+1)\), the ma…

Explicit lower bounds for opaque sets of unit square and unit disc
Markus Kiderlen, Florian Pausinger
https://arxiv.org/abs/2509.08842 https://arxiv.org/pd…

Explicit lower bounds for opaque sets of unit square and unit disc
Explicit lower bounds for the length of the shortest opaque set for the unit disc and the unit square in the Euclidean plane are derived. The results are based on an explicit application of the general method of Kawamura, Moriyama, Otachi and Pach. While the bound for the unit square only slightly improves the existing bound, the bound for the unit disc is new and based on the idea that the free parameters in the general method can be optimized due to the strong symmetry properties of the disc.…

sp_colocation: Social co-locations (2018)
Network of colocations between peoople, based on the information on which RFID readers received information from the RFID tags. Namely, we define two individuals to be in co-presence if the same exact set of readers have received signals from both individuals during a 20s time window.
This network has 332 nodes and 18613039 edges.
Tags: Social, Offline, Unweighted, Weighted, Temporal, Metadata

There may be an $n$-entangled set but no $n 1$-entangled sets
Jorge Antonio Cruz Chapital
https://arxiv.org/abs/2509.01029 https://arxiv.org/pdf/2509.01029…

There may be an $n$-entangled set but no $n+1$-entangled sets
In this paper we show that for every $2\leq n\in \mathbb{N}$, the statement "there is an $n$-entangled set, but there are no $n+1$-entangled sets" is consistent. We also prove some theorems which improve our understanding of entangled sets in relation to construction schemes: (1) The axiom FCA$^Δ$ introduced in \cite{finitizationclubch} implies the existence of $n$-entangled sets which are not $n+1$-entangled. (2) $\mathfrak{m}_\mathcal{F}>ω_1$ implies the non-existence of entangled sets. Thu…

Stable first order theories as simplicial profinite sets
Misha Gavrilovich
https://arxiv.org/abs/2510.00854 https://arxiv.org/pdf/2510.00854

Stable first order theories as simplicial profinite sets
We rewrite simplicially the standard definitions of a complete first order theory, a model of it, and various characterisations of stability of a complete first order theory. In our reformulations the simplicial language replaces the standard definitions based on syntax, making them formally unnecessary. We view a complete first-order theory as a symmetric simplicial object in the category of profinite sets and open continuous maps, defined by the functor sending a finite set of variables int…

Geometry of wave damping on the torus
Kiril Datchev, Perry Kleinhenz, Antoine Prouff
https://arxiv.org/abs/2509.05239 https://arxiv.org/pdf/2509.05239

Geometry of wave damping on the torus
Energy decay rates of damped waves on the torus depend on the behavior of the damping near the undamped region and on the geometry of the damped set. In this paper we refine these geometric considerations, by introducing the concept of order of a glancing undamped point, and estimating decay rates in terms of this order. The proof is based on generalizing an averaging argument due to Sun. We also show that damping sets which attain these improvements are generic among polygons and smooth curves.

Randomized Feasibility-Update Algorithms for Stochastic Variational Inequality Problems
Abhishek Chakraborty, Angelia Nedi\'c
https://arxiv.org/abs/2509.12613 https://

Randomized Feasibility-Update Algorithms for Stochastic Variational Inequality Problems
This paper considers stochastic monotone variational inequalities whose feasible region is the intersection of a (possibly infinite) number of convex functional level sets. A projection-based approach or direct Lagrangian-based techniques for such problems can be computationally expensive if not impossible to implement. To deal with the problem, we consider randomized methods that avoid the projection step on the whole constraint set by employing random feasibility updates. In particular, we pr…

Isolation of non-triangle cycles in graphs
Peter Borg, Dayle Scicluna
https://arxiv.org/abs/2510.08361 https://arxiv.org/pdf/2510.08361

Isolation of non-triangle cycles in graphs
Given a set $\mathcal{F}$ of graphs, we call a copy of a graph in $\mathcal{F}$ an $\mathcal{F}$-graph. The $\mathcal{F}$-isolation number of a graph $G$, denoted by $ι(G, \mathcal{F})$, is the size of a smallest set $D$ of vertices of $G$ such that the closed neighbourhood of $D$ intersects the vertex sets of the $\mathcal{F}$-graphs contained by $G$ (equivalently, $G-N[D]$ contains no $\mathcal{F}$-graph). Let $\mathcal{C}$ be the set of cycles, and let $\mathcal{C}'$ be the set of non-trian…

Checking and producing word attractors
Marie-Pierre B\'eal, Maxime Crochemore, Giuseppe Romana
https://arxiv.org/abs/2509.08503 https://arxiv.org/pdf/2…

Checking and producing word attractors
The article focuses on word (or string) attractors, which are sets of positions related to the text compression efficiency of the underlying word. The article presents two combinatorial algorithms based on Suffix automata or Directed Acyclic Word Graphs. The first algorithm decides in linear time whether a set of positions on the word is an attractor of the word. The second algorithm generates an attractor for a given word in a greedy manner. Although this problem is NP-hard, the algorithm is e…

Explicit formulae and topological descriptions of action-minimizing sets for 2-locally potentials of the XY model
Yuika Kajihara, Shoya Motonaga, Mao Shinoda
https://arxiv.org/abs/2510.02678

Explicit formulae and topological descriptions of action-minimizing sets for 2-locally potentials of the XY model
We consider ergodic optimization of a symbolic dynamics with uncountable alphabet $[0,1]$ for 2-locally constant potentials with the twist condition, and give explicit formulae of the associated Mather set and the Aubry set. Moreover, we investigate the total disconnectedness of the (quotient) Aubry set, in which case the differentiability of the potential function makes a remarkable difference. Although these results imply that the (quotient) Aubry set is small enough, we give a complete chara…

Review of Steiner formulas in Fractal Geometry via Support measures and Complex Dimensions
Goran Radunovi\'c (University of Zagreb, Faculty of Science)
https://arxiv.org/abs/2509.05227

Review of Steiner formulas in Fractal Geometry via Support measures and Complex Dimensions
We review the theoretical framework that establishes a crucial bridge between the general Steiner-type formula of Hug, Last, and Weil and the theory of complex (fractal) dimensions of Lapidus et all. Two novel families of geometric functionals are introduced based on the support measures of the set itself as well as of its parallel sets, respectively. The associated scaling exponents provide new tools for extracting geometric information of the set, beyond its classical fractal dimensions while…

Revisiting Madigan and Mosurski: Collapsibility via Minimal Separators
Pei Heng, Yi Sun, Shiyuan He, Jianhua Guo
https://arxiv.org/abs/2510.09024 https://a…

Revisiting Madigan and Mosurski: Collapsibility via Minimal Separators
Collapsibility provides a principled approach for dimension reduction in contingency tables and graphical models. Madigan and Mosurski (1990) pioneered the study of minimal collapsible sets in decomposable models, but existing algorithms for general graphs remain computationally demanding. We show that a model is collapsible onto a target set precisely when that set contains all minimal separators between its non-adjacent vertices. This insight motivates the Close Minimal Separator Absorption (…

Card Sorting with Fewer Cards and the Same Mental Models? A Re-examination of an Established Practice
Eduard Kuric, Peter Demcak, Matus Krajcovic
https://arxiv.org/abs/2509.03232

Card Sorting with Fewer Cards and the Same Mental Models? A Re-examination of an Established Practice
To keep card sorting with a lot of cards concise, a common strategy for gauging mental models involves presenting participants with fewer randomly selected cards instead of the full set. This is a decades-old practice, but its effects lacked systematic examination. To assess how randomized subsets affect data, we conducted an experiment with 160 participants. We compared results between full and randomized 60\% card sets, then analyzed sample size requirements and the impacts of individual pers…

Divisibility among power GCD and power LCM matrices on certain gcd-closed sets
Jixiang Wan, Guangyan Zhu
https://arxiv.org/abs/2510.04799 https://arxiv.org…

Divisibility among power GCD and power LCM matrices on certain gcd-closed sets
Let $(x, y)$ and $[x, y]$ denote the greatest common divisor and the least common multiple of the integers $x$ and $y$ respectively. We denote by $|T|$ the number of elements of a finite set $T$. Let $a,b$ and $n$ be positive integers and let $S=\{x_1, ..., x_n\}$ be a set of $n$ distinct positive integers. We denote by $(S^a)$ (resp. $[S^a]$) the $n\times n$ matrix whose $(i,j)$-entry is the $a$th power of $(x_i,x_j)$ (resp. $[x_i,x_j]$). For any $x\in S$, define $G_{S}(x):=\{d\in S:…

The set of distances in a Polish metric space
John D. Clemens
https://arxiv.org/abs/2509.01027 https://arxiv.org/pdf/2509.01027

The set of distances in a Polish metric space
We show that a set of non-negative reals is the distance set of a separable complete metric space if and only if it is either countable or is an analytic set which has 0 as a limit point. We also consider spaces with simpler distance sets.

Measuring dissimilarity between convex cones by means of max-min angles
Welington de Oliveira, Valentina Sessa, David Sossa
https://arxiv.org/abs/2511.10483 https://arxiv.org/pdf/2511.10483 https://arxiv.org/html/2511.10483
arXiv:2511.10483v1 Announce Type: new
Abstract: This work introduces a novel dissimilarity measure between two convex cones, based on the max-min angle between them. We demonstrate that this measure is closely related to the Pompeiu-Hausdorff distance, a well-established metric for comparing compact sets. Furthermore, we examine cone configurations where the measure admits simplified or analytic forms. For the specific case of polyhedral cones, a nonconvex cutting-plane method is deployed to compute, at least approximately, the measure between them. Our approach builds on a tailored version of Kelley's cutting-plane algorithm, which involves solving a challenging master program per iteration. When this master program is solved locally, our method yields an angle that satisfies certain necessary optimality conditions of the underlying nonconvex optimization problem yielding the dissimilarity measure between the cones. As an application of the proposed mathematical and algorithmic framework, we address the image-set classification task under limited data conditions, a task that falls within the scope of the \emph{Few-Shot Learning} paradigm. In this context, image sets belonging to the same class are modeled as polyhedral cones, and our dissimilarity measure proves useful for understanding whether two image sets belong to the same class.
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A New Algorithm for Computing Integer Hulls of 2D Polyhedral Sets
Chirantan Mukherjee
https://arxiv.org/abs/2509.09134 https://arxiv.org/pdf/2509.09134

A New Algorithm for Computing Integer Hulls of 2D Polyhedral Sets
The $\texttt{IntegerHull}$ function is part of Maple's $\texttt{PolyhedralSets}$ library, which calculates the integer hull of a given polyhedral set. This algorithm works by translating the supporting hyperplanes of the facets of the input polyhedral set inwards till each hyperplane encounters at least one integer point. The polyhedral set is then divided into smaller regions and a brute force method is applied to find the remaining vertices. There are certain edge case scenarios where the c…

sp_colocation: Social co-locations (2018)
Network of colocations between peoople, based on the information on which RFID readers received information from the RFID tags. Namely, we define two individuals to be in co-presence if the same exact set of readers have received signals from both individuals during a 20s time window.
This network has 332 nodes and 18613039 edges.
Tags: Social, Offline, Unweighted, Weighted, Temporal, Metadata

The nine model category structures on the category of sets
Omar Antol\'in-Camarena, Tobias Barthel
https://arxiv.org/abs/2508.15731 https://arxiv.org/p…

The nine model category structures on the category of sets
We give a proof of the folklore theorem, attributed to Goodwillie, that there are precisely nine model structures on the category $\mathsf{Set}$ of sets. This result is deduced from a complete study of lifting problems and the ensuing classification of all weak factorization systems on $\mathsf{Set}$. Moreover, we determine the Quillen equivalences between these model structures and exhibit an explicit example of equivalent model structures that cannot be realized by a single Quillen adjunction.

On exponential separation of analytic self-conformal sets on the real line
Bal\'azs B\'ar\'any, Istv\'an Kolossv\'ary, Sascha Troscheit
https://arxiv.org/abs/2509.07888

On exponential separation of analytic self-conformal sets on the real line
In a recent article, Rapaport showed that the there is no dimension drop for exponentially separated analytic IFSs on the real line. We show that the set of such exponentially separated IFSs in the space of analytic IFSs contains an open and dense set in the $\mathcal{C}^2$ topology. Moreover, we give a sufficient condition for the IFS to be exponentially separated which allows us to construct explicit examples which are exponentially separated. The key technical tool is the introduction of the…

Finite dominating sets for the refueling station location problem in fleet operations
Moddassir Khan Nayeem, Fuhad Ahmed Opu, Omar Abbaas, Sara Abu-Aridah
https://arxiv.org/abs/2509.11441

Finite dominating sets for the refueling station location problem in fleet operations
This study considers a set of routes used by public transportation vehicles and dedicated distribution fleets in a general network. We aim to optimally locate alternative fuel refueling stations in the network to serve these dedicated routes. Deviations from prescribed routes for refueling purposes are allowed. Unlike most related literature, our approach considers all points in the network as candidate refueling station locations. We derive coverage constraints for any candidate location to se…

The inverse of the star discrepancy of a union of randomly digitally shifted Korobov polynomial lattice point sets depends polynomially on the dimension
Josef Dick, Friedrich Pillichshammer
https://arxiv.org/abs/2509.15877

The inverse of the star discrepancy of a union of randomly digitally shifted Korobov polynomial lattice point sets depends polynomially on the dimension
The star discrepancy is a quantitative measure of the uniformity of a point set in the unit cube. A central quantity of interest is the inverse of the star discrepancy, $N(\varepsilon, s)$, defined as the minimum number of points required to achieve a star discrepancy of at most~$\varepsilon$ in dimension~$s$. It is known that $N(\varepsilon, s)$ depends only linearly on the dimension~$s$. All known proofs of this result are non-constructive. Finding explicit point set constructions that achiev…

Falconer's problem for dot product on paraboloids
Chun-Kai Tseng
https://arxiv.org/abs/2509.11710 https://arxiv.org/pdf/2509.11710

Falconer's problem for dot product on paraboloids
We establish dimensional thresholds for dot product sets associated with compact subsets of translated paraboloids. Specifically, we prove that when the dimension of such a subset exceeds $ \frac{5}{4} = \frac{3}{2} - \frac{1}{4} $ in $\mathbb{R}^3$, and $ \frac{d}{2} - \frac{1}{4} - \frac{1}{8d - 4} $ in $\mathbb{R}^d$ for $d\geq 4$, its dot product set has positive Lebesgue measure. This result demonstrates that if a compact set in $ \mathbb{R}^d $ exhibits a paraboloidal structure, then th…

Convergence exponent of Dirichlet non-improvable numbers in the theory of continued fractions
Xiaoyan Tan, Zhenliang Zhang
https://arxiv.org/abs/2509.23059 https://

Convergence exponent of Dirichlet non-improvable numbers in the theory of continued fractions
Let $x \in [0,1)$ be an irrational number with continued fraction expansion $[a_1(x),a_2(x), \cdots,a_n(x),\cdots]$ and $q_n(x)$ be the denominator of its $n$-th convergent. We establish, for any $α,β$ in $[0,+\infty]$, the Hausdorff dimension formula of the intersections of the sets of Dirichlet non-improvable numbers and the level set of convergent exponent, i.e. $$ G(α,β): =\left\{x\in[0,1)\colon τ(x)=α,\,\,\text{and} \,\, \limsup_{n\to\infty}\frac{\log (a_n(x)a_{n+1}(x))}{\log q_n(x…

sp_colocation: Social co-locations (2018)
Network of colocations between peoople, based on the information on which RFID readers received information from the RFID tags. Namely, we define two individuals to be in co-presence if the same exact set of readers have received signals from both individuals during a 20s time window.
This network has 100 nodes and 394247 edges.
Tags: Social, Offline, Unweighted, Weighted, Temporal, Metadata

Abstraction Principles and the Size of Reality
Bokai Yao
https://arxiv.org/abs/2508.21105 https://arxiv.org/pdf/2508.21105…

Abstraction Principles and the Size of Reality
The Fregean ontology can be naturally interpreted within set theory with urelements, where objects correspond to sets and urelements, and concepts to classes. Consequently, Fregean abstraction principles can be formulated as set-theoretic principles. We investigate how the size of reality-i.e., the number of urelements-interacts with these principles. We show that Basic Law V implies that for some well-ordered cardinal $κ$, there is no set of urelements of size $κ$. Building on recent work by…

Hitting times of shrinking targets: Transversality and an ergodic theorem
Vicente Saavedra-Araya
https://arxiv.org/abs/2510.07450 https://arxiv.org/pdf/251…

Hitting times of shrinking targets: Transversality and an ergodic theorem
In this paper, we investigate ergodic and fractal properties of the sets $$Λ_y:=\Big\{n\in\mathbb{N}:\ \{u_ny\}\in I_n\Big\},$$ where $\{\cdot\}$ denotes the fractional part function, $(u_n)_{n\in\mathbb{N}}$ is an increasing sequence of real numbers, $y\in [0,1]$ and each $I_n$ is a finite union of intervals with decreasing Lebesgue measure. Our main result shows that, under suitable conditions, the set $Λ_y$ is good for pointwise convergence of ergodic averages for Lebesgue almost every $y\…

On Few-Distance Sets in the Plane
Lucas Wang
https://arxiv.org/abs/2510.09800 https://arxiv.org/pdf/2510.09800

On Few-Distance Sets in the Plane
Let $g(k)$ be the maximum size of a planar set that determines at most $k$ distances. We prove $$\fracπ{3\,C(Λ_{hex})}\ k\sqrt{\log k} (1+o(1)) \le g(k) \le C k\log k,$$ so $g(k) \asymp k\sqrt{\log k}$ with an explicit constant from the hexagonal lattice. For any arithmetic lattice $Λ$ we show $$g_Λ(k)\ge (π/4) S^*(Λ) k\sqrt{\log k} (1+o(1)).$$ We also give quantitative stability: unless $X$ is line-heavy or has two popular nonparallel shifts, either almost all ordered pairs lie below a h…

A Composition-Based Approach to EKR Problems
J. B. Ebrahimi, A. Taherkhani
https://arxiv.org/abs/2509.06207 https://arxiv.org/pdf/2509.06207

A Composition-Based Approach to EKR Problems
Let ${\mathcal F}$ be a family of subsets of a finite set $X$. A subfamily of ${\mathcal F}$ is intersecting if any two members of ${\mathcal F}$ has nonempty intersection. We say ${\mathcal F}$ has the EKR property, if, for any fixed element of $X$ the subfamily of all sets in $\mathcal{F}$ that contain $x$ is an intersecting subfamily of $\mathcal{F}$ of possible maximum size. We say ${\mathcal F}$ has the strong EKR property (the strong EKR property), if every intersecting subfamily of $\mat…

Low-Discrepancy Set Post-Processing via Gradient Descent
Fran\c{c}ois Cl\'ement, Linhang Huang, Woorim Lee, Cole Smidt, Braeden Sodt, Xuan Zhang
https://arxiv.org/abs/2511.10496 https://arxiv.org/pdf/2511.10496 https://arxiv.org/html/2511.10496
arXiv:2511.10496v1 Announce Type: new
Abstract: The construction of low-discrepancy sets, used for uniform sampling and numerical integration, has recently seen great improvements based on optimization and machine learning techniques. However, these methods are computationally expensive, often requiring days of computation or access to GPU clusters. We show that simple gradient descent-based techniques allow for comparable results when starting with a reasonably uniform point set. Not only is this method much more efficient and accessible, but it can be applied as post-processing to any low-discrepancy set generation method for a variety of standard discrepancy measures.
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sp_colocation: Social co-locations (2018)
Network of colocations between peoople, based on the information on which RFID readers received information from the RFID tags. Namely, we define two individuals to be in co-presence if the same exact set of readers have received signals from both individuals during a 20s time window.
This network has 100 nodes and 394247 edges.
Tags: Social, Offline, Unweighted, Weighted, Temporal, Metadata

Comparing Left and Right Quotient Sets in Groups
Julian Duvivier, Xiaoyao Huang, Ava Kennon, Say-Yeon Kwon, Steven J. Miller, Arman Rysmakhanov, Pramana Saldin, Ren Watson
https://arxiv.org/abs/2509.00611

Comparing Left and Right Quotient Sets in Groups
For a finite subset $A$ of a group $G$, we define the right quotient set and the left quotient set of $A$, respectively, as $AA^{-1} := \{a_1a_2^{-1}:a_1,a_2\in A\}$, $A^{-1}A := \{a_1^{-1}a_2:a_1,a_2\in A\}$. While the right and left quotient sets are equal if $G$ is abelian, subtleties arise when $G$ is a nonabelian group, where the cardinality difference $|AA^{-1}| - |A^{-1}A|$ may be take on arbitrarily large values. Using the results of Martin and O'Bryant on the cardin…

New constructions and bounds for nonabelian Sidon sets with applications to Tur\'an-type problems
John Byrne, Michael Tait
https://arxiv.org/abs/2509.07750 https://

New constructions and bounds for nonabelian Sidon sets with applications to Turán-type problems
An $S_k$-set in a group $Γ$ is a set $A\subseteqΓ$ such that $α_1\cdotsα_k=β_1\cdotsβ_k$ with $α_i,β_i\in A$ implies $(α_1,\ldots,α_k)=(β_1,\ldots,β_k)$. An $S_k'$-set is a set such that $α_1β_1^{-1}\cdotsα_kβ_k^{-1}=1$ implies that there exists $i$ such that $α_i=β_i\text{ or }β_i=α_{i+1}$. We give explicit constructions of large $S_k$-sets in the group $S_n$ and $S_2$-sets in $S_n\times S_n$ and $A_n\times A_n$. We give probabilistic constructions for `nice' groups which o…

Julia sets appear quasiconformally in the Mandelbrot set, II: A parabolic proof
Tomoki Kawahira, Masashi Kisaka
https://arxiv.org/abs/2510.01013 https://ar…

Julia sets appear quasiconformally in the Mandelbrot set, II: A parabolic proof
Following the ideas of A.~Douady, we give an alternative proof of the authors' result: for any boundary point $c_0$ of the Mandelbrot set $M$, we can find small quasiconformal copies of $M$ in $M$ that are encaged in nested quasiconformal copies of the totally disconnected Julia set of a parameter arbitrarily close to $c_0$.

On the difference of the intersection power graph and the power graph of a finite group
Sudip Bera, Peter J. Cameron
https://arxiv.org/abs/2509.03919 https://

On the difference of the intersection power graph and the power graph of a finite group
The two graphs of the title both have vertex set G. In the intersection power graph, x and y are joined if some non-identity element is a power of both; in the power graph, x and y joined if one is a power of the other. Thus the power graph is a spanning subgraph of the intersection power graph, and we define the edges of the difference graph to be the difference of these edge sets. In this paper, we give a number of results about the difference graph. We examine groups whose power graph and in…

Obstructions for normally spanned sets of vertices
Nicola Lorenz, Max Pitz
https://arxiv.org/abs/2510.04367 https://arxiv.org/pdf/2510.04367

Obstructions for normally spanned sets of vertices
Halin conjectured that a graph has a normal spanning tree if and only if every minor of it has countable colouring number. This has recently been proven by the second author. In this paper, we strengthen this result by establishing the following local version of it: Given a prescribed set of vertices $U$ in a connected graph $G$, there is a normal tree in $G$ that includes $U$ if and only if every $U$-rooted minor of $G$ (i.e. a minor every branch set of which meets $U$) has countable colouri…

Nagumo-Type Characterization of Forward Invariance for Constrained Systems
Olayo Reynaud, Mohamed Maghenem, Adnane Saoud, Sadek Belamfedel Alaoui, Ahmad Hably
https://arxiv.org/abs/2508.20045

Nagumo-Type Characterization of Forward Invariance for Constrained Systems
This paper proposes a Nagumo-type invariance condition for differential inclusions defined on closed constraint sets. More specifically, given a closed set to render forward invariant, the proposed condition restricts the system's dynamics, assumed to be locally Lipschitz, on the boundary of the set restricted to the interior of the constraint set. In particular, when the boundary of the set is entirely within the interior of the constraint set, the proposed condition reduces to the well-known …

Perfect Edge Domination in $P_6$-free Graphs and in Graphs Without Efficient Edge Dominating Sets
Luciano N. Grippo, Min Chih Lin, Camila Vera
https://arxiv.org/abs/2509.04598 h…

Perfect Edge Domination in $P_6$-free Graphs and in Graphs Without Efficient Edge Dominating Sets
An edge of a graph dominates itself along with any edge that shares an endpoint with it. An efficient edge dominating set (also called a dominating induced matching, DIM) is a subset of edges such that each edge of the graph is dominated by exactly one edge in the subset. A perfect edge dominating set is a subset of edges in which every edge outside the subset is dominated by exactly one edge within it. In this article, we establish the NP-completeness of deciding whether a graph that does not …

Random Finite Sumsets and Product Sets in Subsets of the Natural Numbers
Sukrit Chakraborty, Sayan Goswami, Sourav Kanti Patra
https://arxiv.org/abs/2510.01301 https://

Random Finite Sumsets and Product Sets in Subsets of the Natural Numbers
We investigate the occurrence of additive and multiplicative structures in random subsets of the natural numbers. Specifically, for a Bernoulli random subset of $\mathbb{N}$ where each integer is included independently with probability $p\in (0,1)$, we prove that almost surely such a set contains finite sumsets (FS-sets) and finite product sets (FP-sets) of every finite length. In addition, we establish a novel connection between Hindman's partition theorem and the central limit theorem, provid…

Asymptotic structure. V. The coarse Menger conjecture in bounded path-width
Tung Nguyen, Alex Scott, Paul Seymour
https://arxiv.org/abs/2509.08762 https://…

Asymptotic structure. V. The coarse Menger conjecture in bounded path-width
Menger's theorem tells us that if $S,T$ are sets of vertices in a graph $G$, then (for $k\ge0$) either there are $k+1$ vertex-disjoint paths between $S$ and $T$, or there is a set of $k$ vertices separating $S$ and $T$. But what if we want the paths to be far apart, say at distance at least $c$? One might hope that we can find either $k+1$ paths pairwise far apart, or $k$ sets of bounded radius that separate $S$ and $T$, where the bound on the radius is some $\ell$ that depends only on $k,c$ (t…

The odd independence number of graphs, II: Finite and infinite grids and chessboard graphs
Yair Caro, Mirko Petru\v{s}evski, Riste \v{S}krekovski, Zsolt Tuza
https://arxiv.org/abs/2510.01897

The odd independence number of graphs, II: Finite and infinite grids and chessboard graphs
An odd independent set $S$ in a graph $G=(V,E)$ is an independent set of vertices such that, for every vertex $v \in V \setminus S$, either $N(v) \cap S = \emptyset$ or $|N(v) \cap S| \equiv 1$ (mod 2), where $N(v)$ stands for the open neighborhood of $v$. The largest cardinality of odd independent sets of a graph $G$, denoted $α_{od}(G)$, is called the odd independence number of $G$. This new parameter is a natural companion to the recently introduced strong odd chromatic number. …

Iterated sumset expansion in $\mathbb{F}_p^n$
Manik Dhar, Sammy Luo
https://arxiv.org/abs/2510.08857 https://arxiv.org/pdf/2510.08857

Iterated sumset expansion in $\mathbb{F}_p^n$
Given a set $A \subseteq \mathbb{F}_p^n$, what conditions does one need to guarantee that iterated sumsets of the form $A+\cdots+A$ expand quickly (say, within $O(p)$ terms) to the whole space? When only the size of $A$ is known, such expansion results are only possible when $|A|>\frac{1}{p}|\mathbb{F}_p^n|$. However, heuristic considerations suggest that expansion should begin with much smaller sets under just mild ``nondegeneracy'' conditions. In this paper, we confirm thi…

On the partition regularity of arithmetic progressions and linear equations in k-IP-sets
Rapha\"el Giordano
https://arxiv.org/abs/2509.17013 https://a…

On the partition regularity of arithmetic progressions and linear equations in k-IP-sets
In this paper, we provide versions of Van der Waerden's theorem and Rado's theorem for finite colorings of IP-sets and k-IP-sets. Here, by an IP-set we mean a set of integers that contains all finite sums of an infinite subset of N, and we define k-IP-sets similarly to IP-sets but allowing each summand to appear with multiplicity bounded by k - 1.

Constructions of Large m-Distance Sets on Triangular Lattice
Li-Ren Bao, Wei-Hsuan Yu
https://arxiv.org/abs/2509.00880 https://arxiv.org/pdf/2509.00880

Constructions of Large m-Distance Sets on Triangular Lattice
An $m$-distance set is a collection of points such that the distances between any two points have $m$ possible values. We use two different methods to construct large $m$-distance sets on the triangular lattices. One is to use the first m smallest distances and find the largest cliques, and the other is using the notions of hexagons. Multiplicities of the distances were observed for comparison for the two methods.

Discrepancy Beyond Additive Functions with Applications to Fair Division
Alexandros Hollender, Pasin Manurangsi, Raghu Meka, Warut Suksompong
https://arxiv.org/abs/2509.09252 ht…

Discrepancy Beyond Additive Functions with Applications to Fair Division
We consider a setting where we have a ground set $M$ together with real-valued set functions $f_1, \dots, f_n$, and the goal is to partition $M$ into two sets $S_1,S_2$ such that $|f_i(S_1) - f_i(S_2)|$ is small for every $i$. Many results in discrepancy theory can be stated in this form with the functions $f_i$ being additive. In this work, we initiate the study of the unstructured case where $f_i$ is not assumed to be additive. We show that even without the additivity assumption, th…

On upper bounds on the number of parts in the problem of partitioning sets into parts of smaller diameter
Arthur Igorevich Bikeev, Andrei Mikhailovich Raigorodskii
https://arxiv.org/abs/2508.14578

On upper bounds on the number of parts in the problem of partitioning sets into parts of smaller diameter
In the present paper, we study problems related to the classical Borsuk's problem. Recall that the Borsuk's problem consists in finding the smallest number $ f(n) $ of parts of smaller diameter into which an arbitrary set of diameter 1 in Euclidean space $ {\mathbb R}^n $ can be divided. Here we will discuss the quantity $ χ(n,b) $ which differs from the quantity $ f(n) $ in that in its definition an arbitrary set of diameter 1 in $ {\mathbb R}^n $ must be partitioned into parts whose diameter…

Well quasi-order and atomicity for combinatorial structures under consecutive orders
Victoria Ironmonger, Nik Ru\v{s}kuc
https://arxiv.org/abs/2510.01852 https://

Well quasi-order and atomicity for combinatorial structures under consecutive orders
We consider partially ordered sets of combinatorial structures under consecutive orders, meaning that two structures are related when one embeds in the other such that `consecutive' elements remain consecutive in the image. Given such a partially ordered set, we may ask decidability questions about its avoidance sets: subsets defined by a finite number of forbidden substructures. Two such questions ask, given a finite set of structures, whether its avoidance set is well quasi-ordered (i.e. cont…

On the isolation number of graphs with minimum degree four
Wayne Goddard, Michael A. Henning
https://arxiv.org/abs/2508.21551 https://arxiv.org/pdf/2508.21…

On the isolation number of graphs with minimum degree four
An isolating set in a graph $G$ is a set $S$ of vertices such that removing $S$ and its neighborhood leaves no edge. The isolation number $ι(G)$ of $G$ (also known as the vertex-edge domination number) is the minimum size among all isolating sets of $G$. We provide a technique for proving upper bounds on this parameter for graphs with a given minimum degree. For example, we show that if $G$ has order~$n$ and minimum degree at least~$4$, then $ι(G) \le 13n/41$, and if $G$ is also triangle-free…