86 research outputs found
Decidability of definability issues in the theory of real addition
Given a subset of we can associate with every
point a vector space of maximal dimension with the
property that for some ball centered at , the subset coincides inside
the ball with a union of lines parallel with . A point is singular if
has dimension . In an earlier paper we proved that a -definable relation is actually definable in if and only if the number of singular points is finite and every rational
section of is -definable, where a rational section is
a set obtained from by fixing some component to a rational value. Here we
show that we can dispense with the hypothesis of being -definable by assuming that the components of the singular points
are rational numbers. This provides a topological characterization of
first-order definability in the structure . It also
allows us to deliver a self-definable criterion (in Muchnik's terminology) of
- and -definability for a
wide class of relations, which turns into an effective criterion provided that
the corresponding theory is decidable. In particular these results apply to the
class of recognizable relations on reals, and allow us to prove that it is
decidable whether a recognizable relation (of any arity) is
recognizable for every base .Comment: added sections 5 and 6, typos corrected. arXiv admin note: text
overlap with arXiv:2002.0428
Contextual partial commutations
We consider the monoid T with the presentation which is "close" to trace monoids. We prove two different types of results. First, we give a combinatorial description of the lexicographically minimum and maximum representatives of their congruence classes in the free monoid {a; b}* and solve the classical equations, such as commutation and conjugacy in T. Then we study the closure properties of the two subfamilies of the rational subsets of T whose lexicographically minimum and maximum cross-sections respectively, are rational in {a; b}*. © 2010 Discrete Mathematics and Theoretical Computer Science
Contextual partial commutations
We consider the monoid T with the presentation which is "close" to trace monoids. We prove two different types of results. First, we give a combinatorial description of the lexicographically minimum and maximum representatives of their congruence classes in the free monoid {a; b}* and solve the classical equations, such as commutation and conjugacy in T. Then we study the closure properties of the two subfamilies of the rational subsets of T whose lexicographically minimum and maximum cross-sections respectively, are rational in {a; b}*. © 2010 Discrete Mathematics and Theoretical Computer Science
On the Maximum Coefficients of Rational Formal Series in Commuting Variables
Abstract. We study the maximum function of any R+-rational formal series S in two commuting variables, which assigns to every integer n ∈ N, the maximum coefficient of the monomials of degree n. We show that if S is a power of any primitive rational formal series, then its maximum function is of the order Θ(n k/2 λ n ) for some integer k ≥ −1 and some positive real λ. Our analysis is related to the study of limit distributions in pattern statistics. In particular, we prove a general criterion for establishing Gaussian local limit laws for sequences of discrete positive random variables
- …