Subdirectly irreducible medial quandles

We describe all subdirectly irreducible medial quandles. We show that they fall within one of four disjoint classes. In particular, in the finite case they are either connected (and therefore Alexander quandles) or reductive. Moreover, we provide a representation of all non-connected subdirectly irreducible medial quandles.     

On spherical expansions of zonal functions on Euclidean spheres

This paper describes a new approach to the problem of computing spherical expansions of zonal functions on Euclidean spheres. We derive an explicit formula for the coefficients of the expansion expressing them in terms of the Taylor coefficients of the profile function rather than (as done usually) in terms of its integrals against Gegenbauer polynomials. Our proof of this result is based on a polynomial identity equivalent to the canonical decomposition of homogeneous polynomials and uses only basic properties of this decomposition together with simple facts concerning zonal harmonic polynomials. As corollaries, we obtain direct and apparently new derivations of the so-called plane wave expansion and of the expansion of the Poisson kernel for the unit ball. Received: 26 January 2007     

How far can we go with Amitsur’s theorem in differential polynomial rings?

A well-known theorem by S. A. Amitsur shows that the Jacobson radical of the polynomial ring R[x] equals I[x] for some nil ideal I of R. In this paper, however, we show that this is not the case for differential polynomial rings, by proving that there is a ring R which is not nil and a derivation D on R such that the differential polynomial ring R[x;D] is Jacobson radical. We also show that, on the other hand, the Amitsur theorem holds for a differential polynomial ring R[x;D], provided that D is a locally nilpotent derivation and R is an algebra over a field of characteristic p > 0. The main idea of the proof introduces a new way of embedding differential polynomial rings into bigger rings, which we name platinum rings, plus a key part of the proof involves the solution of matrix theory-based problems.     

Generalized entropy in expanded semigroups and in algebras with neutral element

The complex product of (non-empty) subalgebras of a given algebra from a variety \(\mathcal{V}\) is again a subalgebra if and only if the variety \(\mathcal{V}\) has the so-called generalized entropic property. This paper is devoted to algebras with a neutral element or with a semigroup operation. We investigate relationships between the generalized entropic property and the commutativity of the fundamental operations of the algebra. In particular, we characterize the algebras with a neutral element that have the generalized entropic property. Furthermore, we show that, similarly as for n-monoids and n-groups, for inverse semigroups, the generalized entropic property is equivalent to commutativity.     

On some congruences of power algebras

In a natural way we can “lift” any operation defined on a set A to an operation on the set of all non-empty subsets of A and obtain from any algebra (A, Ω) its power algebra of subsets. In this paper we investigate extended power algebras (power algebras of non-empty subsets with one additional semilattice operation) of modes (entropic and idempotent algebras). We describe some congruence relations on these algebras such that their quotients are idempotent. Such congruences determine some class of non-trivial subvarieties of the variety of all semilattice ordered modes (modals).     

Brucine and two solvates

The crystal structures of brucine (2,3‐di­methoxy­strychnidin‐10‐one), C₂₃H₂₆N₂O₄, brucine acetone solvate, C₂₃H₂₆N₂O₄·C₃H₆O, and brucine 2‐propanol solvate dihydrate, C₂₃H₂₆N₂O₄·C₃H₇O·2H₂O, have been determined. Crystals of brucine and its 2‐propanol solvate dihydrate exhibit similar monolayer sheet packing, whereas crystals of the acetone solvate adopt a different mode of packing, as brucine pillars. The solvent appears to control the brucine self‐assembly on the basis of common donor–acceptor properties of the surfaces.     

Phase diagram and dissolution studies of the fenofibrate–acetylsalicylic acid system

Enhancement of the dissolution rate of poorly soluble compounds through the formation of drug–drug eutectics was investigated using fenofibrate and acetylsalicylic acid. Solid–liquid equilibria in the system under study were investigated by differential scanning calorimetry (DSC). The phase diagram for the whole range of compositions was constructed. In addition, existence of a metastable polymorph of fenofibrate has been confirmed. The investigation has revealed that acetylsalicylic acid and fenofibrate form a simple eutectic mixture containing 0.958 mol fraction of fenofibrate at the eutectic point. Dissolution rate improvement of fenofibrate correlated with the phase diagram. The amount of fenofibrate released from the solid dispersions that contained fenofibrate as the eutectic mixture with acetylsalicylic acid was at least threefold higher compared to untreated fenofibrate.