Related papers: Average dimension of fixed point spaces with appli…

Average distance between two points in a unit-volume body $K \subset \mathbb{R}^n$ tends to infinity as $n \to \infty$. However, for two small subsets of volume $\varepsilon > 0$ the situation is different. For unit-volume cubes and…

We prove that a finite von Neumann algebra ${\mathcal A}$ is semisimple if the algebra of affiliated operators ${\mathcal U}$ of ${\mathcal A}$ is semisimple. When ${\mathcal A}$ is not semisimple, we give the upper and lower bounds for the…

For every connected, almost simple linear algebraic group $G\leq\mathrm{GL}_{n}$ over a large enough field $K$, every subvariety $V\subseteq G$, and every finite generating set $A\subseteq G(K)$, we prove a general dimensional bound, that…

Given two rational maps $f,g: \mathbb{P}^1 \to \mathbb{P}^1$ of degree $d$ over $\mathbb{C}$, DeMarco-Krieger-Ye [DKY22] has conjectured that there should be a uniform bound $B = B(d) > 0$ such that either they have at most $B$ common…

We prove two conjectures of E. Khukhro and P. Shumyatsky concerning the Fitting height and insoluble length of finite groups. As a by-product of our methods, we also prove a generalization of a result of Flavell, which itself generalizes…

For a fixed, compactly supported probability measure $\mu$ on the $d$-dimensional space $\mathbb{R}^d$, we consider the problem of minimizing the $p^{\mathrm{th}}$-power average distance functional over all compact, connected $\Sigma…

For a finite group $G,$ $\mathsf{D}(G)$ is defined as the least positive integer $k$ such that for every sequence $S=g_1\bdot g_2\bdot \dotsc \bdot g_k$ of length $k$ over $G$, there exist $1 \le i_1 < i_2 <\cdots < i_m \le k $ such that…

We consider the problem of achieving average consensus in the minimum number of linear iterations on a fixed, undirected graph. We are motivated by the task of deriving lower bounds for consensus protocols and by the so-called "definitive…

Guo and the second author have shown that the closure $[I]$ in the Drury-Arveson space of a homogeneous principal ideal $I$ in $\mathbb{C}[z_1,...,z_n]$ is essentially normal. In this note, the authors extend this result to the closure of…

In a paper published in 2020 in Studia Mathematica, Abrahamsen et al. proved that in the real space $L_1(\mu)$, where $\mu$ is a non-zero $\sigma$-finite (countably additive non-negative) measure, norm-one elements in finite convex…

A design is a finite set of points in a space on which every "simple" functions averages to its global mean. Illustrative examples of simple functions are low-degree polynomials on the Euclidean sphere or on the Hamming cube. We prove lower…

Let $k$ be an algebraically closed field of characteristic $p > 0$, and let $G$ be a simple simply-connected algebraic group over $k$ that is defined and split over the prime field $\mathbb{F}_p$. In this paper we investigate situations…

We give the sharp lower bound of the volume product of $n$-dimensional convex bodies which are invariant under a discrete subgroup $SO(K)=\{ g \in SO(n); g(K)=K \}$, where $K$ is an $n$-cube or $n$-simplex. This provides new partial results…

Given a finite group $G$ and a faithful irreducible $FG$-module $V$ where $F$ has prime order, does $G$ have a regular orbit on $V$? This problem is equivalent to determining which primitive permutation groups of affine type have a base of…

In 1971 Trotter conjectured that every finite poset on at least $3$ points has a pair whose removal does not decrease the dimension by more than $1$. In 1992 Brightwell and Scheinerman introduced fractional dimension of posets, and they…

Let $G=PSL(2,\mathbb{R})$, let $\Gamma$ be a lattice in $G$, and let $\mathcal{H}$ be an irreducible unitary representation of $G$ with square-integrable matrix coefficients. A theorem in [Goodman, de la Harpe, Jones 1989] states that the…

Let $V$ be a finite-dimensional vector space over a finite field, and suppose $G \leq \Gamma \mathrm{L}(V)$ is a group with a unique subnormal quasisimple subgroup $E(G)$ that is absolutely irreducible on $V$. A base for $G$ is a set of…

It is shown that there exists a nontrivial uniform algebra that is Dirichlet on its maximal ideal space and has a dense set of elements that are exponentials. This answers a 65-year-old question of John Wermer and a 17-year-old question of…

We prove a $C^{1,1}$-regularity of minimizers of the functional $$ \int_I \sqrt{1+|Du|^2} + \int_I |u-g|ds,\quad u\in BV(I), $$ provided $I\subset\mathbb{R}$ is a bounded open interval and $\|g\|_\infty$ is sufficiently small, thus…

In this paper, we replace the real numbers by a topological R-module and define R-metric spaces $(X,d)$. Also, we prove some common fixed point theorems on R-module metric spaces. We obtain, as a particular case the Perov theorem.