Related papers: A short proof for tc(K) = 4

The boxicity of a graph G, denoted as box(G) is defined as the minimum integer t such that G is an intersection graph of axis-parallel t-dimensional boxes. A graph G is a k-leaf power if there exists a tree T such that the leaves of the…

We give a new explicit construction for the simplicial group $K(A,n)$. We explain the topological interpretation and discuss some possible applications.

We give a combinatorial description of the embedded contact homology chain complex of the unit cotangent bundle of the Klein bottle with the standard flat Riemannian metric. Using pseudoholomorphic curves coming from the associated…

For a smooth, closed $n$-manifold $M$, we define an upper semi-continuous integer-valued complexity function on $H^1(M;{\mathbb R})$ using Morse theory. This measures how far an integral class is from being a fiber of a fibration. The fact…

We show that if we enrich first order logic by allowing quantification over isomorphisms between definable ordered fields the resulting logic, L(Q_{Of}), is fully compact. In this logic, we can give standard compactness proofs of various…

We prove that there exists a constant $k$ with the property: if $\calC$ is a conjugacy class of a finite group $G$ such that every $k$ elements of $\calC$\ generate a solvable subgroup then $\calC$ generates a solvable subgroup. In…

Given a convex disk $K$ and a positive integer $k$, let $\vartheta_T^k(K)$ and $\vartheta_L^k(K)$ denote the $k$-fold translative covering density and the $k$-fold lattice covering density of $K$, respectively. Let $T$ be a triangle. In a…

In 2012, Bartholdi, Siegenthaler, and Zalesskii computed the rigid kernel of the Hanoi towers group. We present a simpler proof that the rigid kernel is the Klein 4 group. In the course of the proof, we also compute the rigid stabilizers…

We found that Lemma 6.3 in the paper ``The crossing number of $K_{4,n}$ on the torus and the Klein bottle" is wrong.

Distributed systems are critical to reliable and scalable computing; however, they are complicated in nature and prone to bugs. To modularly manage this complexity, network middleware has been traditionally built in layered stacks of…

We study the higher (sequential) topological complexity, a numerical homotopy invariant for the planar polygon spaces. For these spaces with a small genetic codes and dimension $m$, Davis showed that their topological complexity is either…

Complexification, from real connective K-theory to complex connective K-theory, sits in a well-known cofiber sequence between multiplication by eta and a map related to realification. We show how this cofiber sequence factors as one goes up…

Using the concept of real tunneling configurations (classical signature change) and nucleation energy, we explore the consequences of an alternative minimization procedure for the Euclidean action in multiple-dimensional quantum cosmology.…

The subsumption problem with respect to terminologies in the description logic ALC is EXPTIME-complete. We investigate the computational complexity of fragments of this problem by means of allowed Boolean operators. Hereto we make use of…

We generalize a recent result of Clausen: For a number field with integers O, we compute the K-theory of locally compact O-modules. For the rational integers this recovers Clausen's result as a special case. Our method of proof is quite…

Quantified CTL (QCTL) extends the temporal logic CTL with quantifications over atomic propositions. This extension is known to be very expressive: QCTL allows us to express complex properties over Kripke structures (it is as expressive as…

The arithmetic complexity counts the number of algebraically independent entries in the periodic continued fraction $\theta=[b_1,\dots, b_N, \overline{a_1,\dots,a_k}]$. If $\mathscr{A}_{\theta}$ is a noncommutative torus corresponding to…

We denote the number of distinct topologies which can be defined on a set $X$ with $n$ elements by $T(n)$. Similarly, $T_0(n)$ denotes the number of distinct $T_0$ topologies on the set $X$. In the present paper, we prove that for any prime…

We define the topological complexity sequence of a group as the sequence of topological complexities of its Milnor constructions. This sequence may be regarded as an intrinsic refinement of the topological complexity of a group and, unlike…

An \emph{affine subtorus} of the compact torus $T=(S^1)^n$ is a translated copy of a Lie subgroup. Given a finite collection $T_1,\ldots, T_k$ of such subtori, and a prime $p$, we describe an explicit chain complex that calculates the group…