Related papers: Solved and unsolved problems about abelian squares
We recall the Alon-Tarsi conjecture on the number of even latin squares. We introduce a map which switches the parity of a latin square under certain requirements. An example is included.
We show that the epimorphism problem is solvable for targets that are virtually cyclic or a product of an Abelian group and a finite group.
This work connects two mathematical fields - computational complexity and interval linear algebra. It introduces the basic topics of interval linear algebra - regularity and singularity, full column rank, solving a linear system, deciding…
We say that an infinite word w is weak abelian periodic if it can be factorized into finite words with the same frequencies of letters. In the paper we study properties of weak abelian periodicity, its relations with balance and frequency.…
We prove that for any sequence of binary alphabets $\mathcal{A}_1,\mathcal{A}_2,\dots$, there exists a cube-free word $c_1c_2\dots$ so that $c_1\in\mathcal{A}_1,c_2\in\mathcal{A}_2,\dots$. In particular, for every $n$, there are at least…
We develop a new framework of relative algebroids to address existence and classification problems of geometric structures subject to partial differential equations.
The aim of this paper is investigating the existence of solutions of some semilinear elliptic problems on open bounded domains when the nonlinearity is subcritical and asymptotically linear at infinity and there is a perturbation term which…
We characterize the finite sets S of words such that that the iterated shuffle of S is co-finite and we give some bounds on the length of a longest word not in the iterated shuffle of S.
This paper is dedicated to present an exact solution for a nonlinear differential equation so-called Abel equation. This equation was known as one of the group of unsolvable differential equations. The present method is applicable for any…
An example of a cocomplete abelian category that is not complete is constructed.
These notes discuss in an informal manner the construction and some properties of 1- and 2-gerbes. They are mainly based on the author's previous work in this area, which is reviewed here, and to some extent improved upon. The main emphasis…
This is an update on, and expansion of, our paper Open problems on $\beta\omega$ in the book Open Problems in Topology.
In this note, we give the explicit formula for the number of multisubsets of a finite abelian group $G$ with any given size such that the sum is equal to a given element $g\in G$. This also gives the number of partitions of $g$ into a given…
Symmetries and reductions of some algebraic equations are considered. Transformations that preserve the form of several algebraic equations, as well as transformations that reduce the degree of these equations, are described. Illustrative…
There is a long-standing problem of algebra to extend the symmetric monoidal structure of abelian groups, given by the tensor product, to a non abelian setting. In this paper we show that such an extension is possible. Morover our non…
We show that the abelian complexity function of the ordinary paperfolding word is a 2-regular sequence.
In this article we give a survey on open problems and conjectures concerning L^2-invariants. We cover the whole portfolio and not only certain aspects as they are considered in the previous more specialized (and within their scope more…
We study words that barely avoid repetitions, for several senses of "barely". A squarefree (respectively, overlap-free, cubefree) word is irreducible if removing any one of its interior letters creates a square (respectively, overlap,…
It is shown that various questions about the existence of simple closed curves in normal subgroups of surface groups are undecidable.
We review the open problems in the theory of deformations of zero-dimensional objects, such as algebras, modules or tensors. We list both the well-known ones and some new ones that emerge from applications. In view of many advances in…