For compact Hausdorff spaces $X$ and $Y$ we give a partial description of a lattice isomorphism between certain sublattices of continuous functions on $X$ and $Y$ with values in the unit interval $I=[0,1]$. In particular, we give an improvement of Marovt's result concerning order isomorphisms between the lattices of $I$-valued continuous functions on first countable compact Hausdorff spaces $X$ and $Y$.

Let $X$ be a hyperkähler variety admitting a Lagrangian fibration. Beauville's “splitting property” conjecture predicts that fibres of the Lagran-gian fibration should have a particular behaviour in the Chow ring of $X$. We study this conjectural behaviour for two very classical examples of Lagrangian fibrations.

We introduce and study the set-star-$K$-Hurewicz property of a topological space. We investigate its relationship with other covering properties of topological spaces.

We consider crossed homomorphisms between associative algebras. We construct a differential graded Lie algebra whose Maurer-Cartan elements are given by crossed homomorphisms. This allows us to define cohomology for a crossed homomorphism. Finally, we study linear deformations, formal deformations and extendibility of finite order deformations of a crossed homomorphism in terms of the cohomology theory.

The aim of this article is to provide characterizations for subadditivity-like growth conditions for the so-called associated weight functions in terms of the defining weight sequence. Such growth requirements arise frequently in the literature and are standard when dealing with ultradifferentiable function classes defined by Braun-Meise-Taylor weight functions since they imply or even characterize important and desired consequences for the underlying function spaces, e.g. closedness under composition.

Let $L$ be a zero-dimensional frame and $\mathfrak ZL$ be the ring of integer-valued continuous functions on $L$. We associate with each sublocale of $\zeta L$, the Banaschewski compactification of $L$, an ideal of $\mathfrak ZL$, and study the behaviour of these types of ideals. The socle of $\mathfrak ZL$ is shown to be always the zero ideal, in contrast with the fact that the socle of the ring $\mathcal RL$ of real-valued continuous functions of $L$ is not necessarily the zero ideal. The ring $\mathfrak ZL$ has been shown by B. Banaschewski to be (isomorphic to) a subring of $\mathcal RL$, so that ideals of the larger ring can be contracted to the smaller one. We show that the contraction of the socle of $\mathcal RL$ to $\mathfrak ZL$ is the ideal of $\mathfrak ZL$ associated with the join (in the coframe of sublocales of $\zeta L)$ of all nowhere dense sublocales of $\zeta L$. It also appears in other guises.

For Lipschitz test functions we propose a new bound of the solution of the Stein equation related to the generalized inverse Gaussian (resp. the Kummer) distribution. This bound is derived using the general approach established in [4] for distributions satisfying a certain differential equation, and thus is optimal for Lipschitz test functions. The main contribution of this paper is to establish an explicit expression of the bound as a function of the parameters of the distribution in terms of the modified Bessel function of the third kind (resp. the confluent hypergeometric function of the second kind). Under a restriction on the parameters we also obtain an optimal bound for the first derivative of the solution. A recurrence formula is established using the iterative technique [4,5] in order to bound derivatives of any order, for test functions smooth enough.

We prove that every connected locally finite regular graph is either isomorphic to a Schreier graph, or has a double cover which is isomorphic to a Schreier graph.