Related papers: The complex plank problem, revisited

Given $n\geq 2$, $z_{ij}\in\mathbb{T}$ such that $z_{ij}=\overline z_{ji}$ for $1\leq i,j\leq n$ and $z_{ii}=1$ for $1\leq i\leq n$, and integers $p_1,...,p_n\geq 1$, we show that the universal $\mathrm{C}^*$-algebra generated by unitaries…

Kelly's theorem states that a set of $n$ points affinely spanning $\mathbb{C}^3$ must determine at least one ordinary complex line (a line passing through exactly two of the points). Our main theorem shows that such sets determine at least…

Let $V$ be a finite dimensional complex vector space and $W\subset \GL(V)$ be a finite complex reflection group. Let $V^{\reg}$ be the complement in $V$ of the reflecting hyperplanes. A classical conjecture predicts that $V^{\reg}$ is a…

An integral polytope is a polytope whose vertices have integer coordinates. A unimodular triangulation of an integral polytope in $\mathbb{R}^d$ is a triangulation in which all simplices are integral with volume $1/d!$. A classic result of…

A linear mapping upon real n-dimensional space, where the dimension n is odd, has a real eigenvalue-eigenvector pair. The corresponding statement for complex vector spaces holds true for any dimension n, but should be easy to demonstrate…

We consider a generalized angle in complex normed vector spaces. Its definition corresponds to the definition of the well known Euclidean angle in real inner product spaces. Not surprisingly it yields complex values as `angles'. This…

We study cluster algebras arising from cluster tubes. We obtain categorical interpretations for $g$-vectors, $c$-vectors and denominator vectors for cluster algebras of type $\mathrm{C}$ with respect to arbitrary initial seeds. In…

We note that the recent polynomial proofs of the spherical and complex plank covering problems by Zhao and Ortega-Moreno give some general information on zeros of real and complex polynomials restricted to the unit sphere. As a corollary of…

This paper gives a new elementary proof of the theorem that all vector bundles on $\mathbb P^1$ split into the direct sum of line bundles. The proof is based on the study of divisors associated to germs of sections at the generic point.

This paper explores some sufficient conditions for the enhanced solvability of strong vector equilibrium problems, which can be established via a variational approach. Enhanced solvability here means existence of solutions, which are strong…

The aim of this paper is to give an existence result for a class of one-dimensional, non-convex, non-coercive problems in the Calculus of Variations. The main tools for the proof are an existence theorem in the convex case and the closure…

Any collection of $n$ compact convex planar sets $K_1,\dots, K_n$ defines a vector of ${n\choose 2}$ mixed areas $V(K_i,K_j)$ for $1\leq i<j\leq n$. We show that for $n\geq 4$ these numbers satisfy certain Pl\"ucker-type inequalities.…

S. Ekhad and D. Zeilberger recently proved that the multivariate generating function for the number of simple singular vector tuples of a generic $m_1 \times \cdots \times m_d$ tensor has an elegant rational form involving elementary…

For positive integers $i_1,...,i_k$ with $i_1 > 1$, we define the multiple $t$-value $t(i_1,...,i_k)$ as the sum of those terms in the usual infinite series for the multiple zeta value $\zeta(i_1,...,i_k)$ with odd denominators. Like the…

A well-known result says that the Euclidean unit ball is the unique fixed point of the polarity operator. This result implies that if, in $\mathbb{R}^n$, the unit ball of some norm is equal to the unit ball of the dual norm, then the norm…

We show that Kaplansky's unit conjecture is true, under the assumption that the underlying field is complex.

We solve a class of lifting problems involving approximate polynomial relations (soft polynomial relations). Various associated C*-algebras are therefore projective. The technical lemma we need is a new manifestation of Akemann and…

In this work we study arrangements of $k$-dimensional subspaces $V_1,\ldots,V_n \subset \mathbb{C}^\ell$. Our main result shows that, if every pair $V_{a},V_b$ of subspaces is contained in a dependent triple (a triple $V_{a},V_b,V_c$…

We prove that certain vector bundles over surfaces are ample if they are so when restricted to divisors, certain numerical criteria hold, and they are semistable (with respect to $\det(E)$). This result is a higher-rank version of a theorem…

Let $X_{m,d}\subset \mathbb {P}^N$, $N:= \binom{m+d}{m}-1$, be the order $d$ Veronese embedding of $\mathbb {P}^m$. Let $\tau (X_{m,d})\subset \mathbb {P}^N$, be the tangent developable of $X_{m,d}$. For each integer $t \ge 2$ let $\tau…