The *-algebra A1 is defined to be the free unital *-algebra with one generator x. A *-ideal I of A1 is defined to be a C*-ideal if A1/I may be embedded into a C*-algebra. It is proved that if I is a *-ideal of A1 generated by polynomials in A1, then I is a C*-ideal. This is not true for general *-ideals of A1.

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.

Bernstein's famous result, that any non-zero module M over the n-th Weyl algebra An satisfies GKdim(M)≥GKdim(An)/2, does not carry over to arbitrary simple affine algebras, as is shown by an example of McConnell. Bavula introduced the notion of filter dimension of simple algebra to explain this failure. Here, we introduce the faithful dimension of a module, a variant of the filter dimension, to investigate this phenomenon further and to study a revised definition of holonomic modules. We compute the faithful dimension for certain modules over a variant of the McConnell example to illustrate the utility of this new dimension.

Let (R, m) be a Noetherian local reduced ring of positive prime characteristic. We show that if R is ℚ-Gorenstein then the test ideal of R localizes, which extends a result of K. E. Smith. We also show that if Rc, is weakly F-regular and ℚ-Gorenstein, then c has a power which is a completely stable test element. This extends results of Höchster and Huneke.

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 S be a finite semigroup. Consider the set p(S) of all elements of S which can be represented as a product of all the elements of S in some order. It is shown that p(S) is contained in the minimal ideal M of S and intersects each maximal subgroup H of M in essentially the same way. The main result shows that p(S) intersects H in a union of cosets of H′.

The Nottingham group can be described as the group of normalized automorphisms of the ring Fp[[t]], namely, those automorphisms acting trivially on tFp[[t]]/t2Fp[[t]]. In this paper we consider certain proper subgroups of the Nottingham group. We prove that these subgroups are identical to their normalizers and that some of them are isomorphic to the Nottingham group.

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.

Let G be the free product of groups A and B, where |A|≥3 and |B|≥2. We construct faithful, irreducible *-representations for the group algebras ℂ[G] and ℓ1(G). The construction gives a faithful, irreducible representation for F[G] when the field F does not have characteristic 2.

A necessary and sufficient condition for confluence of two poles of a class of meromorphic solutions of the KdV equation is introduced and proved.

We consider the following boundary value problem

where λ > 0 and 1 ≤ p ≤ n – 1 is fixed. The values of λ are characterized so that the boundary value problem has a positive solution. Further, for the case λ = 1 we offer criteria for the existence of two positive solutions of the boundary value problem. Upper and lower bounds for these positive solutions are also established for special cases. Several examples are included to dwell upon the importance of the results obtained.

We construct a family of hyperbolic 3-manifolds whose fundamental groups admit a cyclic presentation. We prove that all these manifolds are cyclic branched coverings of S3 over the knot 52 and we compute their homology groups. Moreover, we show that thecyclic presentations correspond to spines of the manifolds.

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.

Let A be an abelian surface and let |D| be a polarization of type (1,3) on A. If (A,|D|) is not a product of elliptic curves, such a polarization induces a finite morphism Q: A →p2C of degree 6. In this paper we describe the branch locus of Q when A is bielliptic in thesense of K. Hulek and S. H. Weintraub (see [13]), generalizing the results proved by Ch. Birkenhake and H. Lange in [4].

We prove that Finsler metrics on Euclidean domains can be approximated in a certain sense by so-called Finsler-type metrics. As an application we improve upon previous estimates on the fundamental solution of higher order parabolic equations.

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].

A sequential construction of a random spanning tree for the Cayley graph of a finitely generated, countably infinite subsemigroup V of a group G is considered. At stage n, the spanning tree T isapproximated by a finite tree Tn rooted at the identity.The approximation Tn+1 is obtained by connecting edges to the points of V that are not already vertices of Tn but can be obtained from vertices of Tn via multiplication by a random walk step taking values in the generating set of V. This construction leads to a compactification of the semigroup V inwhich a sequence of elements of V that is not eventually constant is convergent if the random geodesic through the spanning tree T that joins the identity to the nth element of the sequence converges in distribution as n→∞. The compactification is identified in a number of examples. Also, it is shown that if h(Tn) and #(Tn) denote, respectively, the height and size of the approximating tree Tn, then there are constants 0<ch≤1 and 0≥c# ≤log2 such that limn→∞ n–1 h(Tn)= ch and limn→∞n–1 log# (Tn)= c# almost surely.

Let G be a finite group of order N and assume that G acts on a Cohen-Macaulay local ring A as automorphisms of rings. Let N be a unit in A. For a given G-stable ideal I in A we denote by R(I) = ⊕n≥0In and = G(I) = ⊕n≥0In/In+1 the Rees algebra and the associated graded ring of I, respectively. Then G naturally acts on R(I) and G(I) too. In this paper the conditions under which the invariant subrings R(I)G of R(I) are Cohen-Macaulay and/or Gorenstein rings are described in connection with the corresponding ring-theoretic properties of G(I)G and the a-invariants a(G(I)G of G(I)G. Consequences and some applications are discussed.

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.