In analogy to ordinary q-additive functions based on q-adic expansions one may use Cantor expansions with a Cantor base Q to define (strongly) Q-additive functions. This paper deals with distribution properties of multi-dimensional sequences which are generated by such Q-additive functions. If in each component we have the same Cantor base Q, then we show that uniform distribution already impli...

A monoid M is a Cale monoid with base Q if for every nonunit x ∈ M there exists a positive integer n such that xn factors uniquely up to order and associates as elements from Q ⊆ M\M. An integral domain D is a Cale domain with base Q, if its multiplicative monoid of nonzero elements is a Cale monoid with base Q. We explore the basic properties of Cale monoids and integral domains. In particular...

The signature of a labelled tree (and hence of its prefix-closed branch language) is the sequence of the degrees of the nodes of the tree in the breadth-first traversal. In a previous work, we have characterised the signatures of the regular languages. Here, the trees and languages that have the simplest possible signatures, namely the periodic ones, are characterised as the sets of representat...

We discuss some challenging open problems in the geometric control theory and sub-Riemannian geometry. It is getting harder to prove theorems and easier to force other people to prove them when you are sixty. Some colleagues asked me to describe interesting open problems in geometric control and sub-Riemannian geometry. Here I list few really challenging problems; some of them are open for a lo...

W develop a new local search algorithm for binary optimization problems, whose complexity and performance are explicitly controlled by a parameter Q, measuring the depth of the local search neighborhood. We show that the algorithm is pseudo-polynomial for general cost vector c, and achieves a w2/42w−15 approximation guarantee for set packing problems with exactly w ones in each column of the co...

It is known that in PG(3, q), q > 19, a partial flock of a quadratic cone with q − ε planes, can be extended to a unique flock if ε < 14 √ q, and a similar and slightly stronger theorem holds for the case q even. In this paper we prove the analogue of this result for cones with base curve of higher degree.

theory of q-difference equations to a comparable degree of completeness. This program includes in particular the complete theory of the convergence and divergence of formal series, the explicit determination of the essential transcendental invariants (constants in the canonical form), the inverse Riemann theory both for the neighborhood of x = o and in the complete plane (case of rational coeff...