Related papers: There are no realizable 15_4- and 16_4-configurati…

We compute all isomorphism classes of simplicial arrangements in the real projective plane with up to 27 lines. It turns out that Gr\"unbaums catalogue is complete up to 27 lines except for four new arrangements with 22, 23, 24, 25 lines,…

One of my recent papers transforms an NP-Complete problem into the question of whether or not a feasible real solution exists to some Linear Program. The unique feature of this Linear Program is that though there is no explicit bound on the…

The "Gr\"unbaum Incidence Calculus" is the common name of a collection of operations introduced by Branko Gr\"unbaum to produce new $(n_{4})$ configurations from various input configurations. In a previous paper, we generalized two of these…

It has been shown by several authors that there exists a non-solvable Bieberbach group of dimension $15$. In this note we show that this is in fact a minimal dimension for such kind of groups.

It is known that the $n$-qubit system has no unextendible product bases (UPBs) of cardinality $2^n-1$, $2^n-2$ and $2^n-3$. On the other hand the $n$-qubit UPBs of cardinality $2^n-4$ exist for all $n\ge3$. We prove that they do not exist…

We call an $\alpha \in \mathbb{R}$ regainingly approximable if there exists a computable nondecreasing sequence $(a_n)_n$ of rational numbers converging to $\alpha$ with $\alpha - a_n < 2^{-n}$ for infinitely many $n \in \mathbb{N}$. We…

We give necessary and sufficient conditions for a sequence to be exactly realizable as the sequence of numbers of periodic points in a dynamical system. Using these conditions, we show that no non-constant polynomial is realizable, and give…

In this paper, we consider Problem 14.44 in the Kourovka notebook, which is a conjecture about the number of conjugacy classes of a finite group. While elementary, this conjecture is still open and appears to elude any straightforward…

Given integers $k_1, k_2$ with $0\le k_1<k_2$, the determinations of all positive integers $q$ for which there exists a perfect Splitter $B[-k_1, k_2](q)$ set is a wide open question in general. In this paper, we obtain new necessary and…

A recurring theme in finite group theory is understanding how the structure of a finite group is determined by the arithmetic properties of group invariants. There are results in the literature determining the structure of finite groups…

We prove that for any $n\geq 4$ there are infinitely many real homotopy types of $2n$-dimensional nilmanifolds admitting generalized complex structures of every type $k$, for $0 \leq k \leq n$. This is in deep contrast to the…

We establish the existence of simple designs with parameters $2$-$(55,10,4)$, $3$-$(20,5,4)$, $3$-$(21,7,30)$, $4$-$(15,5,2)$, $4$-$(16,8,45)$, $5$-$(16,7,10)$, and $5$-$(17,8,40)$, which have previously been unknown. For the corresponding…

We first establish a sufficient condition for an additively idempotent semiring to be nonfinitely based. As applications, we exhibit several examples of additively idempotent semirings satisfying this condition, including a $4$-element…

Five equivalence classes had been found for systems of two second-order ordinary differential equations, transformable to linear equations (linearizable systems) by a change of variables. An "optimal (or simplest) canonical form" of linear…

A facet of an hyperplane arrangement is called external if it belongs to exactly one bounded cell. The set of all external facets forms the envelope of the arrangement. The number of external facets of a simple arrangement defined by $n$…

We say that a loop is unbreakable when it does not have nontrivial subloops. While the cyclic groups of prime order are the only unbreakable finite groups, we show that nonassociative unbreakable loops exist for every order n >= 5. We…

In contrast to finite arithmetic configurations, relatively little is known about which infinite patterns can be found in every set of natural numbers with positive density. Building on recent advances showing infinite sumsets can be found,…

A new infinite family of examples of finite non-bicolorable configurations of rays in Hilbert space is described. Such configurations appear in the analysis of quantum mechanics in terms of Bell's inequalities and Kochen-Specker theorem and…

Consider the set of all natural numbers that are co-prime to primes less than or equal to a given prime. Then given a consecutive pair of numbers in that set with an arbitrary even gap, we prove there exists an unbounded number of actual…

We prove that it is decidable if a finitely based permutation class contains infinitely many simple permutations, and establish an unavoidable substructure result for simple permutations: every sufficiently long simple permutation contains…