The operator that constructs the pseudovariety generated by the idempotent-generated semi-groups of a given pseudovariety is investigated. Several relevant examples of pseudovarieties generated by their idempotent-generated elements are given, as well as some properties of this operator. Particular attention is paid to the pseudovarieties in {J, R, L, DA} concerning this operator and their generator ranks and idempotent-generator ranks.

We provide certain unusual generalizations of Clausen's and Orr's theorems for solutions of fourth-order and fifth-order generalized hypergeometric equations. As an application, we present several examples of algebraic transformations of Calabi–Yau differential equations.

Following some remarks made by O'Grady and Oguiso, the potential density of rational points on the second punctual Hilbert scheme of certain K3 surfaces is proved.

Let , ρ > 2, be the ρ-variation of the heat semigroup associated to the harmonic oscillator H = ½(−Δ + |x|2). We show that if f ∈ L∞ (ℝ), the (f)(x) < ∞, a.e. x ∈ ℝ. However, we find a function G ∈ L∞ (ℝ), such that (G)(x) ∉ L∞ (ℝ). We also analyse the local behaviour in L∞ of the operator . We find that its growth is smaller than that of a standard singular integral operator. As a by-product of our work we obtain an L∞ (ℝ) function F, such that the square function

a.e. x ∈ ℝ, where is the classical Poisson kernal in ℝ.

Poincaré's Polyhedron Theorem is a widely known valuable tool in constructing manifolds endowed with a prescribed geometric structure. It is one of the few criteria providing discreteness of groups of isometries. This work contains a version of Poincaré's Polyhedron Theorem that is applicable to constructing fibre bundles over surfaces and also suits geometries of non-constant curvature. Most conditions of the theorem, being as local as possible, are easy to verify in practice.

Using a non-smooth version (due to Marano and Motreanu) of a variational principle of Ricceri we prove the existence of infinitely many solutions for certain systems of differential inclusions with various types of boundary conditions.

We refine a result of Dubickas on the maximal multiplicity of the roots of a complex polynomial, and obtain several separability criteria for complex polynomials with large leading coefficient. We also give p-adic analogous results for polynomials with integer coefficients.

Among the values of a binary quadratic form, there are many twins of fixed distance. This is shown in quantitative form. For quadratic forms of discriminant −4 or 8 a corresponding result is obtained for triplets.

Two subgroups A and B of a group G are said to be totally completely conditionally permutable (tcc-permutable) in G if X permutes with Yg for some g ∊ 〈X, Y〉, for all X ≤ A and Y ≤ B. We study the belonging of a finite product of tcc-permutable subgroups to a saturated formation of soluble groups containing all finite supersoluble groups.

We consider orientable closed connected 3-manifolds obtained by performing Dehn surgery on the components of some classical links such as Borromean rings and twisted Whitehead links. We find geometric presentations of their fundamental groups and describe many of them as 2-fold branched coverings of the 3-sphere. Finally, we obtain some topological applications on the manifolds given by exceptional surgeries on hyperbolic 2-bridge knots.

Homological properties of several Banach left L1(G)-modules have been studied by Dales and Polyakov and recently by Ramsden. In this paper, we characterize some homological properties of and as Banach left L1(G)-modules, such as flatness, injectivity and projectivity.

Let p be a prime number. For a positive integer n and a p-adic number ξ, let λn(ξ) denote the supremum of the real numbers λ such that there are arbitrarily large positive integers q such that ‖qξ‖p,‖qξ2‖p,…,‖qξn‖p are all less than q−λ−1. Here, ‖x‖p denotes the infimum of |x−n|p as n runs through the integers. We study the set of values taken by the function λn.

A compact contact Ricci soliton (whose potential vector field is the Reeb vector field) is Sasaki–Einstein. A compact contact homogeneous manifold with a Ricci soliton is Sasaki–Einstein.

We define an infinite class of fractals, called horizontally and vertically blocked labyrinth fractals, which are dendrites and special Sierpiński carpets. Between any two points in the fractal there is a unique arc α; the length of α is infinite and the set of points where no tangent to α exists is dense in α.

Let k be a field, let k* = k \ {0} and let C2 be a cyclic group of order 2. We compute all of the braided monoidal structures on the category of k-vector spaces graded by the Klein group C2 × C2. For the monoidal structures we compute the explicit form of the 3-cocycles on C2 × C2 with coefficients in k*, while, for the braided monoidal structures, we compute the explicit form of the abelian 3-cocycles on C2 × C2 with coefficients in k*. In particular, this will allow us to produce examples of quasi-Hopf algebras and weak braided Hopf algebras with underlying vector space k[C2 × C2].

There is an unfortunate error in Theorem 4.1 of our paper. However, the statement of the theorem remains true with a correct construction of adding a tail to enlarge the dynamical system.

A Lehmer number is a composite positive integer n such that ϕ(n)|n − 1. In this paper, we show that given a positive integer g > 1 there are at most finitely many Lehmer numbers which are repunits in base g and they are all effectively computable. Our method is effective and we illustrate it by showing that there is no such Lehmer number when g ∈ [2, 1000].

We study classes of higher-order singular boundary-value problems on a time scale with a positive parameter λ in the differential equations. A homeomorphism and homomorphism ø are involved both in the differential equation and in the boundary conditions. Criteria are obtained for the existence and uniqueness of positive solutions. The dependence of positive solutions on the parameter λ is studied. Applications of our results to special problems are also discussed. Our analysis mainly relies on the mixed monotone operator theory. The results here are new, even in the cases of second-order differential and difference equations.

By making use of Merle's general shooting method we investigate Dirac equations of the form

Here it is possible that F(0) = −∞ and that F(s) defined on (0,+∞) is not monotonously nondecreasing. Our results cover some known ones as a special case.

Based on the work of Adem and Cohen, this note describes an explicit stable decomposition of the space of commuting n-tuples in SU(2) as a wedge of indecomposable summands.