Related papers: MSTD sets and Freiman isomorphisms
What is the higher-dimensional analog of a permutation? If we think of a permutation as given by a permutation matrix, then the following definition suggests itself: A d-dimensional permutation of order n is an [n]^(d+1) array of zeros and…
For a generic degree d smooth map f: N^n -> M^n we introduce its "transverse fundamental group" \pi(f), which reduces to \pi_1(M) in the case where f is a covering, and in general admits a monodromy homomorphism \pi(f) -> S_{|d|};…
We prove the automorphism conjecture for ordered sets of width less than or equal to 11. The proof supports the meta conjecture that a large number of automorphisms is achievable only as some type of product of independent automorphisms on…
Erickson defined the fusible numbers as a set $\mathcal F$ of reals generated by repeated application of the function $\frac{x+y+1}{2}$. Erickson, Nivasch, and Xu showed that $\mathcal F$ is well ordered, with order type $\varepsilon_0$.…
Let $d$ be a positive integer. A finite group is called $d$-maximal if it can be generated by precisely $d$ elements, while its proper subgroups have smaller generating sets. For $d\in\{1,2\}$, the $d$-maximal groups have been classified up…
A dimension group is an ordered abelian group that is an inductive limit of a sequence of simplicial groups, and a stationary dimension group is such an inductive limit in which the homomorphism is the same at every stage. If a simple…
Given integers s,t, define a function phi_{s,t} on the space of all formal series expansions by phi_{s,t} (sum a_n x^n) = sum a_{sn+t} x^n. For each function phi_{s,t}, we determine the collection of all rational functions whose Taylor…
We prove that for an arbitrary upper semi-continuous function $\phi\colon G(1,2) \to [0,1]$ there exists a compact set $F$ in the plane such that $\dim_{\textrm{A}} \pi F = \phi(\pi)$ for all $\pi \in G(1,2)$, where $\pi F$ is the…
Recent work on the use of dimensional reduction for the regularisation of non--supersymmetric theories is reviewed. It is then shown that there exists a class of theories for which a universal form of the soft supersymmetry breaking terms…
We show that there exist constants $\alpha,\epsilon>0$ such that for every positive integer $n$ there is a continuous odd function $f:S^m\to S^n$, with $m\geq \alpha n$, such that the $\epsilon$-expansion of the image of $f$ does not…
Hereditarily finite (HF) set theory provides a standard universe of sets, but with no infinite sets. Its utility is demonstrated through a formalisation of the theory of regular languages and finite automata, including the Myhill-Nerode…
We consider overlap splines that are defined by connecting the patches of piecewise functions via common values at given finite sets of nodes, without using any partitions of the computational domain. It is shown that some classical finite…
We determine the finite groups whose real irreducible representations have different degrees.
Error: Peer-review process exposed an error in Theorem 1 that, unfourtunately, is not repairable. Idempotent semigroups are always finite. See Green and Rees [1952], Siekmann and Szab\'o [1981] for details Anti-unification is a fundamental…
Sidon sets are those sets such that the sums of two of its elements never coincide. They go back to the 30s when Sidon asked for the maximal size of a subset of consecutive integers with that property. This question is now answered in a…
In comparison to graphs, combinatorial methods for the isomorphism problem of finite groups are less developed than algebraic ones. To be able to investigate the descriptive complexity of finite groups and the group isomorphism problem, we…
We provide exact and asymptotic formulae for the number of unrestricted, respectively indecomposable, $d$-dimensional matrices where the sum of all matrix entries with one coordinate fixed equals 2.
A resolving set of a graph is a set of vertices with the property that the list of distances from any vertex to those in the set uniquely identifies that vertex. In this paper, we construct a resolving set of Johnson graphs, doubled Odd…
Given an infinite iterated function system (IFS) $\mathcal{F}$, we define its dimension spectrum $D(\mathcal{F})$ to be the set of real numbers which can be realised as the dimension of some subsystem of $\mathcal{F}$. In the case where…
A finite collection of unit vectors $S \subset \mathbb{R}^n$ is called a spherical two-distance set if there are two numbers $a$ and $b$ such that the inner products of distinct vectors from $S$ are either $a$ or $b$. We prove that if $a\ne…