Tootfinder

On the intersections of nilpotent subgroups in simple groups
Timothy C. Burness, Hong Yi Huang
https://arxiv.org/abs/2508.03479 https://arxiv.org/pdf/2508.…

On the intersections of nilpotent subgroups in simple groups
Let $G$ be a finite group and let $H_p$ be a Sylow $p$-subgroup of $G$. A very recent conjecture of Lisi and Sabatini asserts the existence of an element $x \in G$ such that $H_p \cap H_p^x$ is inclusion-minimal in the set $\{H_p \cap H_p^g \,:\, g \in G\}$ for every prime $p$. This has been proved in several special cases, including all sufficiently large symmetric and alternating groups. For a simple group $G$, in view of a theorem of Mazurov and Zenkov from 1996, the conjecture implies the e…

BMO-Interpolations and Jump Detection for Functions in $BV\cap BMO$
Paz Hashash, Arkady Poliakovsky
https://arxiv.org/abs/2509.04176 https://arxiv.org/pdf/…

BMO-Interpolations and Jump Detection for Functions in $BV\cap BMO$
In this article, we prove interpolation results for $BMO$ in Lorentz spaces. From these results, we derive interpolation theorems in Besov spaces, the space $BV$, and fractional Sobolev spaces. As an application, we obtain geometric information about the jump set of functions in $BV \cap BMO$.

Groups with conjugacy classes of coprime sizes
Rachel D. Camina, Attila Mar\'oti, Emanuele Pacifici, Chris Parker, Kamilla Rekv\'enyi, Jack Saunders, V\'ictor Sotomayor, Gareth Tracey, Martin van Beek
https://arxiv.org/abs/2508.03851

Groups with conjugacy classes of coprime sizes
Suppose that $x$, $y$ are elements of a finite group $G$ lying in conjugacy classes of coprime sizes. We prove that $\langle {x^G} \cap {y^G}\rangle$ is an abelian normal subgroup of $G$ and, as a consequence, that if $x$ and $y$ are $π$-regular elements for some set of primes $π$, then $x^G y^G$ is a $π$-regular conjugacy class in $G$. The latter statement was previously known for $π$-separable groups $G$ and this generalisation permits us to extend several results concerning the common di…

A Refined Kernel for $d$-Hitting Set
Yuxi Liu, Mingyu Xiao
https://arxiv.org/abs/2506.24114 https://arxiv.org/pdf/2506.24114

A Refined Kernel for $d$-Hitting Set
The $d$-Hitting Set problem is a fundamental problem in parameterized complexity, which asks whether a given hypergraph contains a vertex subset $S$ of size at most $k$ that intersects every hyperedge (i.e., $S \cap e \neq \emptyset$ for each hyperedge $e$). The best known kernel for this problem, established by Abu-Khzam [1], has $(2d - 1)k^{d - 1} + k$ vertices. This result has been very widely used in the literature as many problems can be modeled as a special $d$-Hitting Set problem. In thi…

On the Erd\H{o}s-Ko-Rado problem of flags with type $\{1, n-3 \}$ of finite sets
Philipp Heering
https://arxiv.org/abs/2506.20556 https://

On the Erdős-Ko-Rado problem of flags with type $\{1, n-3 \}$ of finite sets
A flag of a finite set $S$ is a set $f$ of non-empty, proper subsets of $S$, such that $X\subseteq Y$ or $Y\subseteq X$ for all $X,Y\in f$. Two flags $f_1$ and $f_2$ of $S$ are opposite if $X_1\cap X_2=\emptyset$, or $X_1\cup X_2=S$ for all $X_1\in f_1$ and $X_2\in f_2$. The set $\{|X| \mid X\in f \}$ is the type of a flag $f$. A set of pairwise non-opposite flags is an Erdős-Ko-Rado set. In 2022 Metsch posed the problem of determining the maximum size of all Erdős-Ko-Rado sets of f…

EU considering tougher Russian oil price cap as bloc aims to pass 18th sanctions package, Bloomberg reports: https://benborges.xyz/2025/07/11/eu-considering-tougher-russian-oil.html

Two results on set families: sturdiness and intersection
Yongjiang Wu, Zhiyi Liu, Lihua Feng, Yongtao Li
https://arxiv.org/abs/2508.19723 https://arxiv.org…

Two results on set families: sturdiness and intersection
This paper resolves two open problems in extremal set theory. For a family $\mathcal{F} \subseteq 2^{[n]}$ and $i, j\in [n]$, we denote $\mathcal{F} (i,\bar{j})=\{F\backslash\{i\}: F\in \mathcal{F}, F\cap\{i,j\}=\{i\}\}$. The sturdiness $β(\mathcal{F})$ is defined as the minimum $|\mathcal{F} (i,\bar{j})|$ over all $i\neq j$. A family $\mathcal{F}$ is called an IU-family if it satisfies the intersection constraint: $F\cap F'\neq \emptyset $ for all $F,F'\in \mathcal{F}$, as well as t…

No Nowhere Continuous Function Maps all Non-Normal Numbers to Normal Numbers
Chokri Manai
https://arxiv.org/abs/2506.15422 https://ar…

No Nowhere Continuous Function Maps all Non-Normal Numbers to Normal Numbers
In this work, we consider the set of non-normal numbers $\mathcal{N}^c$ and ask if there is a non-empty open interval $I$ and a nowhere constant continuous function $ϕ: I \to \rr$ which maps all non-normal numbers to normal numbers, i.e., $φ(I \cap \mathcal{N}^c) \subset \mathcal{N}.$ We answer this question negatively. This result can be seen as a further manifestation of the richness of the null set $\mathcal{N}^c$. Surprisingly, the "bigger" set of normal numbers $\mathcal{N}$ does not sha…

On the intersection of Cantor set with the unit circle and some sequences
Kan Jiang, Derong Kong, Wenxia Li, Zhiqiang Wang
https://arxiv.org/abs/2507.16510 https://

On the intersection of Cantor set with the unit circle and some sequences
For $λ\in(0,1/2)$ let $K_λ$ be the self-similar set in $\mathbb{R}$ generated by the iterated function system $\{f_0(x)=λx, f_1(x)=λx+1-λ\}$. In this paper, we investigate the intersection of the unit circle $\mathbb{S} \subset \mathbb{R}^2$ with the Cartesian product $K_λ \times K_λ$. We prove that for $λ\in(0, 2 - \sqrt{3}]$, the intersection is trivial, i.e., \[ \mathbb{S} \cap (K_λ \times K_λ) = \{(0,1), (1,0)\}. \] If $λ\in [0.330384,1/2)$, then the intersection $\mathbb{S} \cap…

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Browns place Deshaun Watson on physically unable to perform list https://www.nytimes.com/athletic/6509601/2025/07/22/deshaun-watson-browns-pup-list/

Fair coalition in graphs
Saeid Alikhani, Abbas Jafari, Maryam Safazadeh
https://arxiv.org/abs/2507.15080 https://arxiv.org/pdf/2507.1…

Fair coalition in graphs
Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $D\subseteq V$ such that every vertex not in $D$ is adjacent to at least one vertex in $D$. The cardinality of a smallest dominating set of $G$, denoted by $γ(G)$, is the domination number of $G$. For $k \geq 1$, a $k$-fair dominating set ($kFD$-set) in $G$, is a dominating set $S$ such that $|N(v) \cap D|=k$ for every vertex $ v \in V\setminus D$. A fair dominating set in $G$ is a $kFD$-set for some integer $k\geq …

Simpler is Faster: Practical Distance Reporting by Sorting Along a Space-Filling Curve
Sarita de Berg, Ivor van der Hoog, Eva Rotenberg, Emil Toftegaard G{\ae}de
https://arxiv.org/abs/2508.19891

Simpler is Faster: Practical Distance Reporting by Sorting Along a Space-Filling Curve
Range reporting is a classical problem in computational geometry. A (rectangular) reporting data structure stores a point set $P$ of $n$ points, such that, given a (rectangular) query region $Δ$, it returns all points in $P \cap Δ$. A variety of data structures support such queries with differing asymptotic guarantees such as $k$-d trees, range trees, $R$-trees, and quadtrees. A common variant of range queries are distance reporting queries, where the input is a query point $q$ and a radius $…

Wait-free Replicated Data Types and Fair Reconciliation
Petr Kuznetsov, Maxence Perion, Sara Tucci-Piergiovanni
https://arxiv.org/abs/2508.18193 https://ar…

Wait-free Replicated Data Types and Fair Reconciliation
Replication is a standard way to maintain availability of shared data in fault-prone distributed systems. To make sure that the data replicas are up-to-date, they need to synchronize, which typically means engaging the replicas in waiting for coherent responses from each other. The amount of waiting depends on the consistency and availability guarantees we impose on the system. The folklore CAP theory states that strong consistency (the set of replicas create an illusion of one correct server) …

Recent advances in arrow relations and traces of sets
Mingze Li, Jie Ma, Mingyuan Rong
https://arxiv.org/abs/2507.23375 https://arxiv.org/pdf/2507.23375

Recent advances in arrow relations and traces of sets
The arrow relation, a central concept in extremal set theory, captures quantitative relationships between families of sets and their traces. Formally, the arrow relation $(n, m) \rightarrow (a, b)$ signifies that for any family $\mathcal{F} \subseteq 2^{[n]}$ with $|\mathcal{F}| \geqslant m$, there exists an $a$-element subset $T \subseteq [n]$ such that the trace $\mathcal{F}_{|T} = \{ F \cap T : F \in \mathcal{F} \}$ contains at least $b$ distinct sets. This survey highlights r…

On the Exceptional Sets of Transcendental Analytic Functions in Several Variables with Integer Coefficients
Jean Lelis, Bruno De Paula Miranda, Carlos Gustavo Moreira
https://arxiv.org/abs/2506.15914

On the Exceptional Sets of Transcendental Analytic Functions in Several Variables with Integer Coefficients
In 2020, Marques and Moreira proved that every subset of $\overline{\mathbb{Q}} \cap B(0,1)$, which is closed under complex conjugation and contains $0$, is the exceptional set of uncountably many transcendental analytic functions with integer coefficients. In this paper, we extend this result to transcendental analytic functions in several variables. In particular, we show that every subset of $\overline{\mathbb{Q}}^m$ contained in the unit polydisc $Δ(0;1)$, closed under complex conjugation …

Separating subsets from their images
Marco Barbieri, Maru\v{s}a Lek\v{s}e, Primo\v{z} Poto\v{c}nik, Kamilla Rekv\'enyi
https://arxiv.org/abs/2508.20731 https://

Separating subsets from their images
Let $G$ be a transitive permutation group. In this paper, we introduce and study the parameter $\m(G)$, which estimates the size of the largest set of points $A$ such that there exists a permutation $g$ with the property that $A \cap A^g$ is empty. In particular, we focus on deriving general bounds for arbitrary transitive groups, and on the asymptotic behaviour of certain families of primitive groups.

Max-Cut with Multiple Cardinality Constraints
Yury Makarychev, Madhusudhan Reddy Pittu, Ali Vakilian
https://arxiv.org/abs/2507.12607 https://

Max-Cut with Multiple Cardinality Constraints
We study the classic Max-Cut problem under multiple cardinality constraints, which we refer to as the Constrained Max-Cut problem. Given a graph $G=(V, E)$, a partition of the vertices into $c$ disjoint parts $V_1, \ldots, V_c$, and cardinality parameters $k_1, \ldots, k_c$, the goal is to select a set $S \subseteq V$ such that $|S \cap V_i| = k_i$ for each $i \in [c]$, maximizing the total weight of edges crossing $S$ (i.e., edges with exactly one endpoint in $S$). By designing an …

Liar's vertex-edge domination in subclasses of chordal graphs
Debojyoti Bhattacharya, Subhabrata Paul
https://arxiv.org/abs/2507.04721 https://

Liar's vertex-edge domination in subclasses of chordal graphs
Let $G=(V, E)$ be an undirected graph. The set $N_G[x]=\{y\in V|xy\in E\}\cup \{x\}$ is called the closed neighbourhood of a vertex $x\in V$ and for an edge $e=xy\in E$, the closed neighbourhood of $e$ is the set $N_G[x]\cup N_G[y]$, which is denoted by $N_G[e]$ or $N_G[xy]$. A set $L\subseteq V$ is called \emph{liar's vertex-edge dominating set} of a graph $G=(V,E)$ if for every $e_i\in E$, $|N_G[e_i]\cap L|\geq 2$ and for every pair of distinct edges $e_i,e_j\in E$, $|(N_G…

Largest dyadic dual VC-dimension of non-piercing families
Xinqi Huang, Yuzhen Qi, Mingyuan Rong, Zixiang Xu
https://arxiv.org/abs/2506.13606 https://

Largest dyadic dual VC-dimension of non-piercing families
The dyadic dual VC-dimension of a set system \( \mathcal{F} \) is the largest integer \( \ell \) such that there exist \( \ell \) sets \( F_1, F_{2}, \dots, F_\ell \in \mathcal{F} \), where every pair \( \{i, j\} \in \binom{[\ell]}{2} \) is witnessed by an element \( a_{i,j} \in F_i \cap F_j \) that does not belong to any other set \( F_k \) with \( k \in [\ell] \setminus \{i, j\} \). In this paper, we determine the largest dyadic dual VC-dimension of a non-piercing family is exactly $4$, provi…

On denominators of consecutive $\operatorname{SL}(2,{\mathbb N})$-saturated Farey fractions
Jack Anderson, Florin P. Boca, Cristian Cobeli, Alexandru Zaharescu
https://arxiv.org/abs/2508.07951

On denominators of consecutive $\operatorname{SL}(2,{\mathbb N})$-saturated Farey fractions
The sequences of $\operatorname{SL}(2,{\mathbb N})$-saturated Farey fractions were defined in our previous work as ${\mathscr S}_Q := \{ a/q \in {\mathbb Q} \cap (0,1]: q+a+\bar{a} \le Q\}$, where $\bar{a}$ is the multiplicative inverse of $a\pmod{q}$ in $[1,q)$. Here, we prove that the set of $Q$-scaled denominators of consecutive fractions in ${\mathscr S}_Q$ is dense in the region ${\mathcal V}:=\{ (x,y)\in [0,1]^2 : \max \{ (1-3x)/2,2x-1\} \le y \le \max \{ x,1-x\} \}$, and provide a formul…

On orbit sets generated by semigroups of one-dimensional affine functions
Karim F. Shamazov, Alexey L. Talambutsa
https://arxiv.org/abs/2507.06875 https://…

On orbit sets generated by semigroups of one-dimensional affine functions
The one-dimensional orbit set $\langle F : s \rangle$ is formed by the images of a number $s$ under the action of a semigroup generated by the integer affine functions $f_i=a_i x+b_i$ taken from the set $F=\{f_1,\ldots,f_n\}$. P.Erdös established an upper bound $O(x^{σ+ε})$ for the growth function $|\langle F : s \rangle\cap[0,x]|$, where $1/a_1^σ+1/a_2^σ+\ldots + 1/a_n^σ=1$ and $\varepsilon>0$, which was extended to orbit multisets and real affine functions by J.Lagarias. We co…