We present a family of radical convolution Banach algebras on intervals (0,a] which are of Sobolev type; that is, they are defined in terms of derivatives. Among other properties, it is shown that all epimorphisms and derivations of such algebras are bounded. Also, we give examples of nontrivial concrete derivations.

In 1903 Mirimanoff conjectured that Cauchy–Mirimanoff polynomials En are irreducible over ℚ for odd prime n. Polynomials Rn, Sn, Tn are introduced, closely related to En. It is proved that Rm, Sm, Tm are irreducible over ℚ for odd m≥3 , and En, Rn, Sn are irreducible over ℚ, for n=2qm, q=1,2,3,4,5 , and m≥1 odd.

This paper investigates a class of metrics that can be introduced on the set consisting of the union of continuous functions defined on different intervals with values in a fixed metric space, where the union ranges over a family of intervals. Its definition is motivated by the Skorohod metric(s) on càdlàg functions. We show what is essential in transferring the ideas employed in the latter metric to our setting and obtain a general construction for metrics in our case. Next, we define the metric space where elements are sequences of functions from the above mentioned set. We provide conditions that ensure separability and completeness of the constructed metric spaces.

We consider the wreath product of two permutation groups G≤Sym Γ and H≤Sym Δ as a permutation group acting on the set Π of functions from Δ to Γ. Such groups play an important role in the O’Nan–Scott theory of permutation groups and they also arise as automorphism groups of graph products and codes. Let X be a subgroup of Sym Γ≀Sym Δ. Our main result is that, in a suitable conjugate of X, the subgroup of SymΓ induced by a stabiliser of a coordinate δ∈Δ only depends on the orbit of δ under the induced action of X on Δ. Hence, if X is transitive on Δ, then X can be embedded into the wreath product of the permutation group induced by the stabiliser Xδ on Γ and the permutation group induced by X on Δ. We use this result to describe the case where X is intransitive on Δ and offer an application to error-correcting codes in Hamming graphs.

We give a systematic treatment of caloric measure for arbitrary open sets. The caloric measure is defined only on the essential boundary of the set. Our main result gives criteria for the resolutivity of essential boundary functions, and their integral representation in terms of caloric measure. We also characterize the caloric measure null sets in terms of the boundary singularities of nonnegative supertemperatures.