We are going to investigate translation invariant derivations on Lp spaces of locally compact abelian groups, 1[les ]p<∞. By these we mean densely defined closed linear operators which commute with translations and obey a Leibniz rule, that is, T(fg) =(Tf)·g+f·(Tg); see Definition 1 for details.

The original motivation for studying these operators was to find an abstract description of constant coefficient partial differential operators as a link to perturbation theory which is usually formulated in terms of abstract operator theoretic notions. This fits, for example, Schrödinger operator theory (as in [1]). Here, in view of the applications to perturbation theory, the point is that this identification is not just a formal one but includes assertions about domains (Theorem 1).

In this paper, however, we shall concentrate on groups other than ℝn.

This paper is devoted to the study of different types of twisting points of conformal maps. We define the sets of gyration, spiral and oscillation points and we prove, in the case that f is conformal almost nowhere, that the above sets have Hausdorff dimension one. Also we define points of bounded radial oscillation. It is proved that there are always points of π-bounded radial oscillation but there exists a conformal map without points of small bounded radial oscillation.

A univalent harmonic map of the unit disk Δ[ratio ]={z∈[Copf ][ratio ][mid ]z[mid ]<1} is a complex-valued function f(z) on Δ that satisfies Laplace's equation fzz[bar]=0 and is injective. The Jacobian J[ratio ]=[mid ]fz[mid ]2 −[mid ]fz[bar][mid ]2 of a univalent harmonic map can never vanish [18], and so we might as well assume that J>0 throughout Δ. Then [mid ]fz[mid ]>0 and a short computation verifies that the analytic dilatation ω[ratio ] =f[bar]z[bar]/fz is indeed an analytic function, with [mid ]ω[mid ]<1 since J>0. Clearly ω≡0 when f is a conformal map, and in general the dilatation ω measures how far f is from being conformal. Also, if ω happens to be the square of an analytic function, then f ‘lifts’ to give an isothermal coordinate map for a minimal surface, and in that case i/√ω equals the stereographic projection of the Gauss map of the surface.

In this paper we continue our study of the Canonical Element Conjecture (henceforth C.E.C.) via the dualizing complex. Throughout the work (A, m, k) will denote a noetherian complete local ring A of dimension n, m its maximal ideal and k=A/m. Since A is complete, we can find a complete local Gorenstein ring (R, mR, k) (complete intersection) such that dim R=dim A and A=R/I. Let Ω denote the canonical module of A, that is, Ω=HomR (A, R), which may be identified with the annihilator of I in R, an ideal of R. When A is a domain, we change notation and denote I by P; in this case P is a height 0 prime ideal of R.

This work was motivated in part by the following general question: given an ideal I in a Cohen–Macaulay (abbreviated to CM) local ring R such that dim R/I=0, what information about I and its associated graded ring can be obtained from the Hilbert function and Hilbert polynomial of I? By the Hilbert (or Hilbert–Samuel) function of I, we mean the function HI(n) =λ(R/In) for all n[ges ]1, where λ denotes length. Samuel [23] showed that for large values of n, the function HI(n) coincides with a polynomial PI(n) of degree d=dim R. This polynomial is referred to as the Hilbert, or Hilbert–Samuel, polynomial of I. The Hilbert polynomial is often written in the form

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where e0(I), [ctdot ], ed(I) are integers uniquely determined by I. These integers are known as the Hilbert coefficients of I.

This paper brings together two apparently unrelated results about locally compact groups G by giving them a common proof. The first concerns the number of topologically left invariant means on L∞(G), while the second states that the topological centre of the largest semigroup compactification of G is simply G itself. On the way, we introduce as vital tools some new compactifications of the half line ([0, ∞), +), we produce a right invariant pseudometric on a compactly generated G for which the bounded sets are precisely the relatively compact sets, and we receive striking confirmation that some algebraic properties of semigroups can be transferred by maps which are quite far from being homomorphisms.

Let G be a group endowed with its profinite topology, then G is called product separable if the profinite topology of G is Hausdorff and, whenever H1, H2, [ctdot ], Hn are finitely generated subgroups of G, then the product subset H1H2 [ctdot ] Hn is closed in G. In this paper, we prove that if G=F×Z is the direct product of a free group and an infinite cyclic group, then G is product separable. As a consequence, we obtain the result that if G is a generalized free product of two cyclic groups amalgamating a common subgroup, then G is also product separable. These results generalize the theorems of M. Hall Jr. (who proved the conclusion in the case of n=1, [3]), and L. Ribes and P. Zalesskii (who proved the conclusion in the case of that G is a finite extension of a free group, [6]).

In this paper we are concerned with abstract homomorphisms from one algebraic group to another. That is, group homomorphisms where there is no assumption of rationality. Let K be an infinite perfect field and L an algebraically closed field of the same characteristic. Throughout G(K) will denote a Chevalley group of universal type over K.

A group is accessible if there is a G-tree T such that every edge stabilizer is finite and every vertex stabilizer has at most one end. A group is inaccessible if it is not accessible. In [6] I began the study of group actions on protrees. These actions arise naturally when considering inaccessible groups. This study is continued in this paper.

In 1940 Nisnevič published the following theorem [3]. Let (Gα)α∈Λ be a family of groups indexed by some set Λ and (Fα)α∈Λ a family of fields of the same characteristic p[ges ]0. If for each α the group Gα has a faithful representation of degree n over Fα then the free product *α∈ΛGα has a faithful representation of degree n+1 over some field of characteristic p. In [6] Wehrfritz extended this idea. If (Gα)α∈Λ [les ]GL(n, F) is a family of subgroups for which there exists Z[les ]GL(n, F) such that for all α the intersection Gα∩F.1n=Z, then the free product of the groups *ZGα with Z amalgamated via the identity map is isomorphic to a linear group of degree n over some purely transcendental extension of F.

Initially, the purpose of this paper was to generalize these results from the linear to the skew-linear case, that is, to groups isomorphic to subgroups of GL(n, Dα) where the Dα are division rings. In fact, many of the results can be generalized to rings which, although not necessarily commutative, contain no zero-divisors. We have the following.

Let G be a finite group and k be an algebraically closed field of prime characteristic. Corresponding to each closed homogeneous subvariety W of the maximal ideal spectrum of H*(G, k) we construct (usually infinite-dimensional) kG-modules E(W) and F(W) which are idempotent in the sense that E(W) and F(W) are isomorphic (up to projective summands) to E(W) [otimes ] E(W) and F(W) [otimes ] F(W) respectively. We study the properties of these modules, and as an application we use them to describe natural direct sum decompositions of modules in quotient categories.

Throughout ℤp and ℚp will, respectively, denote the ring of p-adic integers and the field of p-adic numbers (for p prime). We denote by [Copf ]p the completion of the algebraic closure of ℚp with respect to the p-adic metric. Let vp denote the p-adic valuation of [Copf ]p normalised so that vp(p)=1. Put [ ]p ={ω∈[Copf ]p[mid ] ωpn=1 for some n[ges ]0} so that [ ]p is the union of cyclic (multiplicative) groups Cpn of order pn (for n[ges ]0).

Let UD(ℤp) denote the [Copf ]p-algebra of all uniformly differentiable functions f[ratio ]ℤp→[Copf ]p under pointwise addition and convolution multiplication *, where for f, g∈UD(ℤp) and z∈ℤp we have

formula here

the summation being restricted to i, j with vp(i+j−z)[ges ]n.

This situation is a starting point for p-adic Fourier analysis on ℤp, the analogy with the classical (complex) theory being substantially complicated by the absence of a p-adic valued Haar measure on ℤp (see [5, 6] for further details).

Let θ be an irrational number in [0, 1] and Aθ the corresponding irrational rotation C*-algebra. Let Aut (Aθ) be the group of all automorphisms of Aθ and Int (Aθ) the normal subgroup of Aut (Aθ) of all inner automorphisms of Aθ. Let Pic (Aθ) be the Picard group of Aθ. In the present note we shall show that if θ is not quadratic, then Pic (Aθ)≅Aut (Aθ)/Int (Aθ) and that if θ is quadratic, then Pic (Aθ) is isomorphic to a semidirect product of Aut (Aθ)/Int (Aθ) with ℤ. Furthermore, in the last section we shall discuss Picard groups of certain Cuntz algebras.

In this paper, we study certain polynomial invariants of links (singular or non-singular) that are related to the Homfly polynomial and Vassiliev's invariants. The Homfly polynomial HL [3] (also known as the Flypmoth polynomial) satisfies the well-known skein relation xHL+ +yHL− +zHL0=0. The Vassiliev invariants [1, 2] (of order 1) satisfy the relations VL× =VL+ −VL− and VL××=0. The invariants that we study satisfy the skein relations

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