where ∆u(k) = u(k+1)−u(k) is the forward difference operator, Z∗ := {k ∈ Z : k 6= 0} and a, α, δ, f, g are functions to be defined later. Difference equations can be seen as a discrete counterpart of PDEs and are usually studied in connection with numerical analysis. In this way, the main operator in Problem (1.1) −∆(a(k − 1,∆u(k − 1))) can be seen as a discrete counterpart of the anisotropic o...

We investigate convergence rates of Tikhonov regularization for nonlinear ill-posed problems when both the right-hand side and the operator are corrupted by noise. Two models of operator noise are considered, namely uniform noise bounds and point-wise noise bounds. We derive convergence rates for both noise models in Hilbert and in Banach spaces. These results extend existing results where the ...

We present sharp bounds on the supremum norm of DjSh for j ≥ 2, where D is the differential operator and S the Stein operator for the standard normal distribution. The same method is used to give analogous bounds for the exponential, Poisson and geometric distributions, with D replaced by the forward difference operator in the discrete case. We also discuss applications of these bounds to the c...

A set A of positive integers is called a coprime Diophantine powerset if the shifted product ab + 1 of two different elements a and b of A is always a pure power, and the occurring pure powers are all coprime. We prove that each coprime Diophantine powerset A ⊂ {1, . . . , N} has |A| 8000 log N/ log log N for sufficiently large N. The proof combines results from extremal graph theory with numbe...

Automata over infinite words provide a powerful framework, which we can use to solve various decision problems. However, the automatized reasoning with restricted classes of automata over infinite words is often simpler and more efficient. For instance, weak deterministic Büchi automata, which recognize the ω-regular languages in the Borel class Fσ ∩Gδ, can be handled algorithmically almost as ...

Timed automata are a widely studied model. Its decidability has been proved using the so-called region automaton construction. This construction provides a correct abstraction for the behaviours of timed automata, but it suffers from a state explosion and is thus not used in practice. Instead, algorithms based on the notion of zones are implemented using adapted data structures like DBMs. When ...

Timed automata are a widely studied model. Its decidability has been proved using the so-called region automaton construction. This construction provides a correct abstraction for the behaviours of timed automata, but it does not support a natural implementation and, in practice, algorithms based on the notion of zones are implemented using adapted data structures like DBMs. When we focus on fo...

In this paper a binary relation between L-sets, called attachment, is defined by means of a suitable family of completely coprime filters in L. Examples are given using several kinds of lattice-ordered algebras and a class of attachments is determined whose elements generalize Pu-Liu’s quasi-coincidence relation. Mapping any L-set on X to the set of the attached L-points one gets a frame map fr...

We present here a view on abstraction based on the relation between sentences in a partially ordered language L and truth values of these sentences on a set of instances W . In Formal Concept Analysis, this relation is materialized as a lattice denoted as G that relates L and the powerset P(W )). We show here that projections on a lattice (here either L or the powerset P(W )) that are known to ...