Related papers: Which DNR can be minimal

Given an unbounded domain $\Omega$ of a Hadamard manifold $M$, it makes sense to consider the problem of finding minimal graphs with prescribed continuous data on its cone-topology-boundary, i.e., on its ordinary boundary together with its…

We prove a form of the $\cos \pi \rho$ theorem which gives strong estimates for the minimum modulus of a transcendental entire function of order zero. We also prove a generalisation of a result of Hinkkanen that gives a sufficient condition…

We show that for any odd $k$ and any instance of the Max-kXOR constraint satisfaction problem, there is an efficient algorithm that finds an assignment satisfying at least a $\frac{1}{2} + \Omega(1/\sqrt{D})$ fraction of constraints, where…

We introduce higher order variants of the Yang-Mills functional that involve $(n-2)$th order derivatives of the curvature. We prove coercivity and smoothness of critical points in Uhlenbeck gauge in dimensions $\mathrm{dim}M\le 2n$. These…

We prove that every smooth closed manifold admits a smooth real-valued function with only two critical values. We call a function of this type a \emph{Reeb function}. We prove that for a Reeb function we can prescribe the set of minima (or…

One of the fundamental results in graph minor theory is that for every planar graph $H$, there is a minimum integer $f(H)$ such that graphs with no minor isomorphic to $H$ have treewidth at most $f(H)$. A lower bound for ${f(H)}$ can be…

Let $k$ and $n$ be natural numbers. Let $\omega_k(n)$ denote the number of distinct prime factors of $n$ with multiplicity $k$ as studied by Elma and the third author. We obtain asymptotic estimates for the first and the second moments of…

Minimum dominating set is a basic local covering problem and a core task in distributed computing. Despite extensive study, in the classic LOCAL model there exist significant gaps between known algorithms and lower bounds. Chang and Li…

Let G be a finite group. A subgroup M of G is said to be an NR-subgroup if, whenever K is normal in M, then K^G\cap M=K, where K^G is the normal closure of K in G. Using the Classification of Finite Simple Groups, we prove that if every…

In this paper, we show that while almost all functions require exponential size branching programs to compute, for all functions $f$ there is a branching program computing a doubly exponential number of copies of $f$ which has linear size…

We examine the computable part of the differentiability hierarchy defined by Kechris and Woodin. In that hierarchy, the rank of a differentiable function is an ordinal less than omega_1 which measures how complex it is to verify…

Hajnal and Szemeredi proved that every graph G with |G|=ks and minimum degree at least k(s-1) contains k vertex disjoint s-cliques; moreover this degree bound is optimal. We extend their theorem to directed graphs by showing that every…

As part of the graph minor project, Robertson and Seymour showed in 1990 that the class of graphs that can be embedded in a given surface can be characterized by a finite set of minimal excluded minors. However, their proof, because…

For a set $A\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let $R_A(n)$ denote the number of ordered pairs $(a,a')\in A\times A$ such that $a+a'=n$. The celebrated Erd\H{o}s-Tur\'an conjecture says that, if $R_A(n)\ge 1$ for all sufficiently…

In 2017, Andrews, Dixit, Schultz and Yee introduced the function $\overline{\textrm{spt}}_\omega(n)$, which denotes the number of smallest parts in the overpartitions of $n$ in which the smallest part is always overlined and all odd parts…

We consider a magnetic Schr\"odinger operator $H^h=(-ih\nabla-\vec{A})^2$ with the Dirichlet boundary conditions in an open set $\Omega \subset {\mathbb R}^3$, where $h>0$ is a small parameter. We suppose that the minimal value $b_0$ of the…

We prove that for every planar graph $X$ of treedepth $h$, there exists a positive integer $c$ such that for every $X$-minor-free graph $G$, there exists a graph $H$ of treewidth at most $f(h)$ such that $G$ is isomorphic to a subgraph of…

For a normalized root system $R$ in $\mathbb R^N$ and a multiplicity function $k\geq 0$ let $\mathbf N=N+\sum_{\alpha \in R} k(\alpha)$. Denote by $dw(\mathbf x)=\prod_{\alpha\in R}|\langle \mathbf x,\alpha\rangle|^{k(\alpha)}\, d\mathbf x…

A graph $G$ contains a graph $H$ as an induced minor if $H$ can be obtained from $G$ by vertex deletions and edge contractions. The class of $H$-induced-minor-free graphs generalizes the class of $H$-minor-free graphs, but unlike…

Let $M$ be a closed spin manifold and let $N$ be a closed manifold. For maps $f\colon M\to N$ and Riemannian metrics $g$ on $M$ and $h$ on $N$, we consider the Dirac operator $D^f_{g,h}$ of the twisted Dirac bundle $\Sigma…