We introduce a class of polynomials which induce a permutation on the set of polynomials in one variable of degree less than m over a finite field. We call then Am-permutation polynomials. We also give three criteria to characterize such polynomials.

We give a multidirectional mean value inequality with second order information. This result extends the classical Clarke-Ledyaev's inequality to the second order. As application, we give the uniqueness of viscosity solution of second order Hamilton-Jacobi equations in finite dimensions.

Let F be a field of characteristic p and G a group containing at least one element of order p. It is proved that the group of units of the group algebra FG is a bounded Engel group if and only if FG is a bounded Engel algebra, and that this is the case if and only if G is nilpotent and has a normal subgroup H such that both the factor group G/H and the commutator subgroup H′ are finite p–groups.

Let K be a field of characteristic 0, G the direct sum of two copies of the additive group of integers. For a total order ≺ on G, which is compatible with the addition, and for any ċ1, ċ2 ∈ K, we define G-graded highest weight modules M(ċ1, ċ2, ≺) over the Virasoro-like algebra , indexed by G. It is natural to call them Verma modules. In the present paper, the irreducibility of M (ċ1, ċ2, ≺) is completely determined and the structure of reducible module M (ċ1, ċ2, ≺)is also described.

Let M be a reductive algebraic monoid with zero and unit group G. We obtain a description of the submonoid generated by the idempotents of M. In particular, we find necessary and sufficient conditions for M\G to be idempotent generated.

We construct the freest idempotent-pure expansion of an inverse semigroup, generalizing an expansion of Margolis and Meakin for the group case. We also generalize the Birget-Rhodes prefix expansion to inverse semigroups with an application to partial actions of inverse semigroups. In the process of generalizing the latter expansion, we are led to a new class of idempotent-pure homomorphisms which we term F-morphisms. These play the same role in the theory of idempotent-pure homomorphisms that F-inverse monoids play in the theory of E-unitary inverse semigroups.

Let (X, ρ, μ)d, θ be a space of homogeneous type with d < 0 and θ ∈ (0, 1], b be a para-accretive function, ε ∈ (0, θ], ∣s∣ > ∈ and a0 ∈ (0, 1) be some constant depending on d, ∈ and s. The authors introduce the Besov space bBspq (X) with a0 > p ≧ ∞, and the Triebel-Lizorkin space bFspq (X) with a0 > p > ∞ and a0 > q ≧∞ by first establishing a Plancherel-Pôlya-type inequality. Moreover, the authors establish the frame and the Littlewood-Paley function characterizations of these spaces. Furthermore, the authors introduce the new Besov space b−1 Bs (X) and the Triebel-Lizorkin space b−1 Fspq (X). The relations among these spaces and the known Hardy-type spaces are presented. As applications, the authors also establish some real interpolation theorems, embedding theorems, T b theorems, and the lifting property by introducing some new Riesz operators of these spaces.

A semiprime segment of a ring R is a pair P2 ⊂ P1 of semiprime ideals of R such that ∩ In ⊆ P2 for all ideals I of R with P2 ⊂ I ⊂ P1. In this paper semiprime segments with P1 a comparizer ideal are classified as either simple, exceptional, or archimedean, extending to several classes of rings a classification known for right chain rings. These three types of semiprime segments are also characterized in terms of the pseudo-radical.

We introduce complex differential geometry twisted by a real line bundle. This provides a new approach to understand the various real objects that are associated with an anti-holomorphic involution. We also generalize a result of Greenleaf about real analytic sheaves from dimension 2 to higher dimensions.