A semigroup B in which every element is an idempotent can be embedded into such a semigroup B′, where all the local submonoids are isomorphic, and in such a way that B and B′ satisfy the same equational identities. In view of the properties preserved under this embedding, a corresponding embedding theorem is obtained for regular semigroups whose idempotents form a subsemigroup.

In 2001, Enochs's celebrated flat cover conjecture was finally proven, and the proofs (two different proofs were presented in the same paper) have since generated a great deal of interest among researchers. The results have been recast in a number of other categories and, in particular, for additive categories. In 2008, Mahmoudi and Renshaw considered a similar problem for acts over monoids but used a slightly different definition of cover. They proved that, in general, their definition was not equivalent to that of Enochs, except in the projective case, and left open a number of questions regarding the ‘other’ definition. This ‘other’ definition is the subject of the present paper and we attempt to emulate some of Enochs's work for the category of acts over monoids, and concentrate, in the main, on strongly flat acts. We hope to extend this work to other classes of acts, such as injective, torsion free, divisible and free, in a future report.

We prove that if f is an integrable function on the unit sphere S in ℝn, g is its symmetric decreasing rearrangement and u, v are the harmonic extensions of f, g in the unit ball , then v has larger convex integral means over each sphere rS, 0 < r < 1, than u has. We also prove that if u is harmonic in with |u| < 1 and u(0) = 0, then the convex integral mean of u on each sphere rS is dominated by that of U, which is the harmonic function with boundary values 1 on the right hemisphere and −1 on the left one.

Let X be a smooth irreducible projective curve of genus g and gonality 4. We show that the canonical model of X is contained in a uniquely defined surface, ruled by conics, whose geometry is deeply related to that of X. This surface allows us to define four invariants of X and, hence, to stratify the moduli space of four-gonal curves by means of closed irreducible subvarieties, whose dimensions we compute.

The aim of this paper is to prove the existence of a positive solution for a quasi-linear elliptic problem involving the (p, q)-Laplacian and a convection term, which means an expression that is not in the principal part and depends on the solution and its gradient. The solution is constructed through an approximating process based on gradient bounds and regularity up to the boundary. The positivity of the solution is shown by applying a new comparison principle, which is established here.

For stable degree 0 operations, and also for additive unstable operations of bidegree (0, 0), it is known that the centre of the ring of operations for complex cobordism is isomorphic to the corresponding ring of connective complex K-theory operations. Similarly, the centre of the ring of BP operations is the corresponding ring for the Adams summand of p-local connective complex K-theory. Here we show that, in the additive unstable context, this result holds with BP replaced by BP〈n⌰ for any n. Thus, for all chromatic heights, the only central operations are those coming from K-theory.

When can two fibrewise maps be deformed in a fibrewise fashion until they are coincidence free? In order to get a thorough understanding of this problem (and, more generally, of minimum numbers that are closely related to it) we study the strength of natural geometric obstructions, such as ω-invariants and Nielsen numbers, as well as the related Nielsen theory.

In the setting of sphere bundles, a certain degree map degB turns out to play a decisive role. In many explicit cases it also yields good descriptions of the set ℱ of fibrewise homotopy classes of fibrewise maps. We introduce an addition on ℱ, which is not always single valued but still very helpful. Furthermore, normal bordism Gysin sequences and (iterated) Freudenthal suspensions play a crucial role.

In this work we provide effective bounds and classification results for rational ℚ-factorial Fano varieties with a complexity-one torus action and Picard number 1 depending on the two invariants dimension and Picard index. This complements earlier work by Hausen et al., where the case of a free divisor class group of rank 1 was treated.

In this paper quasilinear elliptic boundary value equations without an Ambrosetti and Rabinowitz growth condition are considered. Existence of a non-trivial solution result is established. For this, we show the existence of a Cerami sequence by using a variant of the mountain-pass theorem due to Schechter. The novelty here is that we may consider nonlinearities that satisfy a local p-superlinear condition and may change sign.

Let M be a closed simply connected 2n-dimensional manifold. The paper is concerned with the cohomology of classifying spaces of connected groups of homeomorphisms of M.

In this paper the existence of infinitely many solutions for a class of Kirchhoff-type problems involving the p-Laplacian, with p > 1, is established. By using variational methods, we determine unbounded real intervals of parameters such that the problems treated admit either an unbounded sequence of weak solutions, provided that the nonlinearity has a suitable behaviour at ∞, or a pairwise distinct sequence of weak solutions that strongly converges to 0 if a similar behaviour occurs at 0. Some comparisons with several results in the literature are pointed out. The last part of the work is devoted to the autonomous elliptic Dirichlet problem.

In this paper we review the classification of isolated quotient singularities over the field of complex numbers due to Zassenhaus, Vincent and Wolf. As an application, we describe Gorenstein isolated quotient singularities over ℂ, generalizing a result of Kurano and Nishi.