Bivariate Koornwinder–Sobolev Orthogonal Polynomials

The so-called Koornwinder bivariate orthogonal polynomials are generated by means of a non-trivial procedure involving two families of univariate orthogonal polynomials and a function $$\rho (t)$$ ρ ( t ) such that $$\rho (t)^2$$ ρ ( t ) 2 is a polynomial of degree less than or equal to 2. In this paper, we extend the Koornwinder method to the case when one of the univariate families is orthogonal with respect to a Sobolev inner product. Therefore, we study the new Sobolev bivariate families obtaining relations between the classical original Koornwinder polynomials and the Sobolev one, deducing recursive methods in order to compute the coefficients. The case when one of the univariate families is classical is analysed. Finally, some useful examples are given.

Revisiting the Factorization of x n + 1 over Finite Fields with Applications

The polynomial x n + 1 over finite fields has been of interest due to its applications in the study of negacyclic codes over finite fields. In this paper, a rigorous treatment of the factorization of x n + 1 over finite fields is given as well as its applications. Explicit and recursive methods for factorizing x n + 1 over finite fields are provided together with the enumeration formula. As applications, some families of negacyclic codes are revisited with more clear and simpler forms.

Limit group invariants for non-free Cantor actions

A Cantor action is a minimal equicontinuous action of a countably generated group $G$ on a Cantor space $X$ . Such actions are also called generalized odometers in the literature. In this work, we introduce two new conjugacy invariants for Cantor actions, the stabilizer limit group and the centralizer limit group. An action is wild if the stabilizer limit group is an increasing sequence of stabilizer groups without bound and otherwise is said to be stable if this group chain is bounded. For Cantor actions by a finitely generated group $G$ , we prove that stable actions satisfy a rigidity principle and furthermore show that the wild property is an invariant of the continuous orbit equivalence class of the action. A Cantor action is said to be dynamically wild if it is wild and the centralizer limit group is a proper subgroup of the stabilizer limit group. This property is also a conjugacy invariant and we show that a Cantor action with a non-Hausdorff element must be dynamically wild. We then give examples of wild Cantor actions with non-Hausdorff elements, using recursive methods from geometric group theory to define actions on the boundaries of trees.