This paper is based on the interpretation of the ring of additive polynomials in one variable over a finite field Fq, as a maximal R-order inside a certain skew-field D, R being a principal ideal domain isomorphic to Fp[T]. The classical (1930's) structure theory of maximal orders in global fields is used to solve enumeration questions involving the iteration of members of Pages from .

In this paper we study the problem of algebraic reflexivity of the isometry group of some important Banach spaces. Because of the previous work in similar topics, our main interest lies in the von Neumann – Schatten p-classes of compact operators. The ideas developed there can be used in ℓp-spaces, Banach spaces of continuous functions and spin factors as well. Moreover, we attempt to attract the attention to this problem from general Banach spaces geometry view-point. This study, we believe, would provide nice geometrical results.

We give an algorithm which for a given set of generators can decide whether a stem product of infinite cyclic groups is a one-relator group. We also generalize this to the case of one-relator products of groups.

We study the relationship between the dual of the K-space defined by means of a polygon and the J-space generated by the dual N-tuple. The results complete the research started in [4]. Special attention is paid to the case when the N-tuple is formed by Banach lattices

In this note we show that if we have an exact sequence of AH algebras (AH stands for “approximately homogeneous”) 0 → I → A → B → 0, then A has the ideal property (i.e., any ideal is generated by its projections) if and only if I and B have the ideal property. Also, we prove that an extension of two AT algebras (AT stands for “approximately circle”) with the ideal property is an AT algebra with the ideal property if and only if the extension is quasidiagonal.

We prove two types of vanishing results for the Euler characteristic.

In this paper, finite travelling waves for the semilinear parabolic systems

are studied, where di > 0, ei > 0, mij ≥ 0 for all 1 ≤ i,j ≤ n, and for all 1 ≤ i ≤ n. Let M = (mij)n × n and A = I – M. It will be proved that (*) has finite travelling waves if and only if all principal minors of A are positive. Moreover, some asymptotic behaviours of finite travelling waves will be obtained.

We study the semicentre of a group algebra K[G] where K is a field of characteristic zero and G is a polycyclic-by-finite group suchthat Δ(G) is torsion-free abelian. Several properties about the structure of this ring are proved, in particular as to when is the semicentre a UFD. Examples are constructed when this is not the case. We also prove necessary and sufficient conditions for every normal element of K[G] which belongs to K[Δ(G)] to be the product of a unit and a semi-invariant.

Let S be a finite semigroup, A be a given subset of S and L, R, H, D and J be Green's equivalence relations. The problem of determining whether there exists a supersemigroup T of S from the class of all semigroups or from the class of finite semigroups, such that A lies in an L or R-class of T has a simple and well known solution (see for example [7], [8] or [3]). The analogous problem for J or D relations is trivial if T is of arbitrary size, but undecidable if T is required to be finite [4] (even if we restrict ourselves to the case |A| = 2 [6]). We show that for the relation H, the corresponding problem is undecidable in both the class of finite semigroups (answering Problem 1 of [9]) and in the class of all semigroups, extending related results obtained by M. V. Sapir in [9]. An infinite semigroup with a subset never lying in a H-class of any embedding semigroup is known and, in [9], the existence of a finite semigroup with this property is established. We present two eight element examples of such semigroups as well as other examples satisfying related properties.

Let K be an algebraic number field with ring of integers OK and f(X) ∈ OK[X]. In this paper we establish improved explicit upper bounds for the size of solutions in OK, of diophantine equations Y2 = f(X), where f(X) has at least three roots of odd order, and Ym = f(X), where m is an integer ≥ 3 and f(X) has at least two roots of order prime to m.

We obtain optimal L2-lower bounds for nonzero solutions to – ΔΨ + VΔ = EΨ in Rn, n ≥ 2, E ∈ R where V is a measurable complex-valued potential with V(x) = 0(|x|-c) as |x|→∞, for some ε∈ R. We show that if 3δ = max{0, 1 – 2ε} and exp (τ|x|1+δ)Ψ ∈ L2(Rn)for all τ > 0, then Ψ; has compact support. This result is new for 0 < ε ½ and generalizes similar results obtained by Meshkov for = 0, and by Froese, Herbst, M. Hoffmann-Ostenhof, and T. Hoffmann-Ostenhof for both ε≤O and ε≥½. These L2-lower bounds are well known to be optimal for ε ≥ ½ while for ε < ½ this last is only known for ε = O in view of an example of Meshkov. We generalize Meshkov's example for ε< ½ and thus show that for complex-valued potentials our result is optimal for all ε ∈ R.

We consider associative algebras filtered by the additive monoid ℕp. We prove that, under quite general conditions, the study of Gelfand-Kirillov dimension of modules over a multi-filtered algebra R can be reduced to the associated ℕp-graded algebra G(R). As a consequence, we show the exactness of the Gelfand-Kirillov dimension when the multi-filtration is finite-dimensional and G(R) is a finitely generated noetherian algebra. Our methods apply to examples like iterated Ore extensions with arbitrary derivations and “homothetic” automorphisms (e.g. quantum matrices, quantum Weyl algebras) and the quantum enveloping algebra of sl(v + 1)

We prove in this paper that if (T, G, ∂) is a perfect and aspherical (Ker ∂ = 1) crossed module, then it admits a universal central extension, whose kernel is the invariant H2(T, G, ∂), that we introduced in [9].

In a recent generalization of the Bernstein polynomials, the approximated function f is evaluated at points spaced at intervals which are in geometric progression on [0, 1], instead of at equally spaced points. For each positive integer n, this replaces the single polynomial Bnf by a one-parameter family of polynomials , where 0 < q ≤ 1. This paper summarizes briefly the previously known results concerning these generalized Bernstein polynomials and gives new results concerning when f is a monomial. The main results of the paper are obtained by using the concept of total positivity. It is shown that if f is increasing then is increasing, and if f is convex then is convex, generalizing well known results when q = 1. It is also shown that if f is convex then, for any positive integer n This supplements the well known classical result that when f is convex.

Banach modules over C*-algebras (von Neumann algebras) that can be represented isometrically as operator modules (normal operator modules, respectively) are characterised.