Related papers: Solved and unsolved problems about abelian squares

An abstract indefinite least squares problem with a quadratic constraint is considered. This is a quadratic programming problem with one quadratic equality constraint, where neither the objective nor the constraint are convex functions.…

In this work, we offer a historical stroll through the vast topic of binary quadratic forms. We begin with a quick review of their history and then an overview of contemporary algebraic developments on the subject.

We consider three aspects of avoiding large squares in infinite binary words. First, we construct an infinite binary word avoiding both cubes xxx and squares yy with |y| >= 4; our construction is somewhat simpler than the original…

In this paper we give a novel solution to a classical completion problem for square matrices. This problem was studied by many authors through time, and it is completely solved in [2, 3]. In this paper we relate this classical problem to a…

In this paper we state some conjectures about q-Fibonacci polynomials which for q=1 reduce to well-known results about Fibonacci numbers and Fibonacci polynomials.

We propose a list of open problems in numerical semigroups.

This brief note gives a survey on results relating to existence of closed points on schemes, including an elementary topological characterization of the schemes with (at least one) closed point.

We present some variations on some of the main open problems on character degrees. We collect some of the methods that have proven to be very useful to work on these problems. These methods are also useful to solve certain problems on zeros…

Several older and more recent results on the boundaries of solvability and unsolvability in tag systems are surveyed. Emphasis will be put on the significance of computer experiments in research on very small tag systems.

In this short note we recall the definition of intrinsically harmonic forms, some known results and some open problems.

Some constructions and bounds on the sizes of semiovals contained in the Hermitian curve are given. A construction of an infinite family of 2-blocking sets of the Hermitian curve is also presented.

Our main result is that any real cubic algebraic number has a continued fraction expansion with polynomial coefficients. Some generalizations are mentioned.

In this short note we present some remarks and conjectures on two of Erd\"os's open problems in number theory.

We study the equality problem for infinite words obtained by iterating morphisms. In particular, we give a practical algorithm to decide whether or not two words generated by primitive morphisms are equal.

We consider a special type of self-similar sets, called fractal squares, and give a brief review on recent results and unsolved issues with an emphasis on their topological properties.

We settle the existence of certain "anti-magic" cubes using combinatorial block designs and graph decompositions to align a handful of small examples.

Some special solutions to the multidimensional Lam\'e and Bourlet type equations are constructed in an explicit form.

In this work we study weighted total least squares problems on infinite dimensional spaces. We show that in most cases this problem does not admit a solution (except in the trivial case) and then, we consider a regularization on the…

We present here a large collection of harmonic and quadratic harmonic sums, that can be useful in applied questions, e.g., probabilistic ones. We find closed-form formulae, that we were not able to locate in the literature.

Not any nonsingular equation over a metabelian group has solution in a larger metabelian group. However, any nonsingular equation over a solvable group with a subnormal series with abelian torsion-free quotients has a solution in a larger…