Related papers: Solved and unsolved problems about abelian squares
We survey classical and recent results on exponents of Diophantine approximation. We give only a few proofs and highlight several open problems.
We show that any abelian variety that is not affine has a nontrivial strongly abelian subvariety. In later papers in this sequence we apply this result to the study of minimal abelian varieties.
We discuss the well-known open problems: the local Steiness problem and the union problem.
The paper deals with locally bounded solutions of a Schilling's problem.
In the first part of this survey, we present classical notions arising in combinatorics on words: growth function of a language, complexity function of an infinite word, pattern avoidance, periodicity and uniform recurrence. Our…
We present a list of open questions in the theory of holomorphic foliations, possibly with singularities. Some problems have been around for a while, others are very accessible.
Here we give a short survey of our new results. References to the complete proofs can be found in the text of this article and in the litterature.
In recent literature there are an increasing number of papers where the forbidden sets of difference equations are computed. We review and complete different attempts to describe the forbidden set and propose new perspectives for further…
We consider so-called squaring the square-puzzles where a given square (or rectangle) should be dissected into smaller squares. For a specific instance of such problems we demonstrate that a mathematically rigorous solution can be quite…
In connection to the development of the field of Combinatorics on Words, we present a list of open problems and conjectures that were stated during the ten last meetings WORDS. We wish to continually update the present document by adding…
We answer affirmatively the open problem proposed by Cabr\'e and Tan in their paper "Positive solutions of nonlinear problems involving the square root of the Laplacian" (see Adv. Math. {\bf 224} (2010), no. 5, 2052-2093).
In this article, we solve some word equations originated from discrete dynamical systems related to antisymmetric cubic map. These equations emerge when we work with primitive and greatest words. We conclude with some applications
In this paper we study the maximal pattern complexity of infinite words up to Abelian equivalence. We compute a lower bound for the Abelian maximal pattern complexity of infinite words which are both recurrent and aperiodic by projection.…
Criteria are given for determining whether an irreducible sextic equation with rational coefficients is algebraically solvable over the complex numbers.
We survey results about computational complexity of the word problem in groups, Dehn functions of groups and related problems.
We study short intervals which contain an ``almost square'', an integer $n$ that can be factored as $n = ab$ with $a$, $b$ close to $\sqrt{n}$. This is related to the problem on distribution of $n^2 \alpha \pmod 1$ and the problem on gaps…
We give an introduction to the study of algebraic hypersurfaces, focusing on the problem of when two hypersurfaces are isomorphic or close to being isomorphic. Working with hypersurfaces and emphasizing examples makes it possible to discuss…
The solution of one Zamfiresku's problem was obtained. We discuss the unsolved questions related to the Mizel's problem.
We solve some decision problems for timed automata which were recently raised by S. Tripakis in [ Folk Theorems on the Determinization and Minimization of Timed Automata, in the Proceedings of the International Workshop FORMATS'2003, LNCS,…
We investigate the problem of the maximum number of cubic subwords (of the form $www$) in a given word. We also consider square subwords (of the form $ww$). The problem of the maximum number of squares in a word is not well understood.…