We prove that for any non-degenerate dendrite D, there exist topologically mixing maps $F : D \to D$ and $f : [0, 1] \to [0, 1]$ such that the natural extensions (as known as shift homeomorphisms) $\sigma _F$ and $\sigma _f$ are conjugate, and consequently the corresponding inverse limits are homeomorphic. Moreover, the map f does not depend on the dendrite D and can be selected so that the inverse limit $\underleftarrow {\lim } (D,F)$ is homeomorphic to the pseudo-arc. The result extends to any finite number of dendrites. Our work is motivated by, but independent of, the recent result of the first and third author on conjugation of Lozi and Hénon maps to natural extensions of dendrite maps.

In neuroleptic long-term medication, only part of the patients accept regular intake of neuroleptic drugs. The question is whether an interval medication regimen as opposed to continuous medication can help to reduce drop outs in patients with critical attitudes towards long-term medication. In a 2-year prospective study, 122 patients were randomised to an interval and 164 to a continuous neuroleptic medication regimen. The drop out rates were 62.5% in the interval and 53.7% in the continuous medication group. Drop outs generally show more negative attitudes towards treatment. Patients with negative attitudes do not do better under interval medication. Moreover, this regimen even requires more cooperation and trust in terms of the necessity of medication on the part of the patient compared to the continuous medication regimen. Interval medication therefore is a strategy which can only be successful in highly cooperative, but not in treatment-reluctant patients.

We consider a one-parameter family of dynamical systems $W:[0,1]\rightarrow [0,1]$ constructed from a pair of monotone increasing diffeomorphisms $W_{i}$ such that $W_{i}^{-1}:$$[0,1]\rightarrow [0,1]$$(i=0,1)$. We characterise the set of symbolic itineraries of $W$ using an attractor $\overline{\unicode[STIX]{x1D6FA}}$ of an iterated closed relation, in the terminology of McGehee, and prove that there is a member of the family for which $\overline{\unicode[STIX]{x1D6FA}}$ is symmetrical.

Let $P$ be a finite $\text{N}$-free poset. We consider the hypergraph $H\left( P \right)$ whose vertices are the elements of $P$ and whose edges are the maximal intervals of $P$. We study the dual König property of $H\left( P \right)$ in two subclasses of $\text{N}$-free class.

Completely regular semigroups CR are regarded here as algebras with multiplication and the unary operation of inversion. Their lattice of varieties is denoted by L(CR). Let B denote the variety of bands and L(B) the lattice of its subvarieties. The mapping V → V ∩ B is a complete homomorphism of L(CR) onto L(B). The congruence induced by it has classes that are intervals, say VB = [VB, VB] for V ∈ L(CR). Here VB = V ∩ B. We characterize VB in several ways, the principal one being an inductive way of constructing bases for v-irreducible band varieties. We term the latter canonical. We perform a similar analysis for the intersection of these varieties with the varieties BG, OBG and B.

In this note we characterize the abelian groups G which have two different proper subgroups N and M such that the subgroup lattice L(G)=[0, M]∪ [N, G] is the union of these intervals.

The joint distributions of the exit time and exit value of a spectrally positive process, from semi-infinite and finite intervals, are derived in the form of Fourier-Laplace transforms. Also the probability that such a process makes its first exit from a finite interval via the lower end point is obtained explicitly.