The main results of the paper are the following: 1. Every locally finite affine complete variety admits a near unanimity term; 2. A locally finite congruence distributive variety is affine complete if and only if all its algebras with no proper subalgebras are affine complete and the variety is generated by one of such algebras. The first of these results sharpens a result of McKenzie asserting that all locally finite affine complete varieties are congruence distributive. The second one generalizes the result by Kaarli and Pixley that characterizes arithmetical affine complete varieties.

We restrict the metaplectic representation to subgroups G of the symplectic group associated to equivariant holomorphic maps into the Siegel disc. We describe the invariant subspaces of the decomposition, and reduce the problem to the decomposition of a space of ‘harmonic’ polynomials under the action of the maximal compact subgroup of G.

An inhomogeneous version of a general form of the Jarník-Besicovitch Theorem is proved.

The Mellin-Stieltjes convolution and related decomposition of distributions in M(α) (the class of distributions μ on (0, ∞) with slowly varying αth truncated moments ) are investigated. Maller shows that if X and Y are independent non-negative random variables with distributions μ and v, respectively, and both μ and v are in D2, the domain attraction of Gaussian distribution, then the distribution of the product XY (that is, the Mellin-Stieltjes convolution μ ^ v of μ and v) also belongs to it. He conjectures that, conversely, if μ ∘ v belongs to D2, then both μ and v are in it. It is shown that this conjecture is not true: there exist distributions μ ∈ D2 and v μ ∈ D2 such that μ ^ v belongs to D2. Some subclasses of D2 are given with the property that if μ ^ v belongs to it, then both μ and v are in D2.

In this paper, we give multihomogeneous estimates for the group of relations linking multiplicatively dependent algebraic numbers. In the process, we raise a question in the style of Lehmer's problem, concerning multidimensional covolumes in the lattice of units. The proofs are based on the Brill-Gordan duality theorem on orthogonal lattices, and the paper closes with an algebraic version of the theorem, concerning orthogonal abelian subvarieties of an arbitrarily polarized abelian variety.

In this note, we propose an extension of the compactness property for Kähler-Einstein metrics to critical metrics of Weyl functional on compact Kähler surfaces.

We generalize dual reductive pairs by using reductive groups that are not necessarily subgroups of symplectic groups and construct the corresponding theta-series liftings for certain types of automorphic forms. We also discuss connections of such generalized theta-series liftings with families of abelian varieties parametrized by an arithmetic variety.

As a consequence of general principles, we add to the array of ‘hulls’ in the category Arch (of archimedean ℓ-groups with ℓ-homomorphisms) and in its non-full subcategory W (whose objects have distinguished weak order unit, whose morphisms preserve the unit). The following discussion refers to either Arch or W. Let α be an infinite cardinal number or ∞, let Homα; denote the class of α-complete homomorphisms, and let R be a full epireflective subcategory with reflections denoted rG: G → rG. Then for each G, there is rαG ∈ Homα (G, R) such that for each ϕ ∈ Homα (G, R), there is unique with . Moreover if every rG is an essential embedding, then, for every α and every G, rαG = rG, and every Homα. If and R consists of all epicomplete objects, then every Homw1. For α = ∞, and for any R, every Hom∞.

We solve the following problem which was posed by Barnes in 1962. For which abelian groups G and H of the same prime power order is it possible to embed the subgroup lattice of G in that of H? It follows from Barnes' results and a theorem of Herrmann and Huhn that if there exists such an embedding and G contains three independent elements of order p2, then G and H are isomorphic. This reduces the problem to the case that G is the direct product of cyclic p-groups only two of which have order larger than p. We determine all groups H for which the desired embedding exists.