Our goal in this paper is to determine conditions on a potential V which ensure that an operator such as

formula here

acting on L2(RN) defines a semigroup in Lp(RN) for various values of p including p=1. The operator is defined as a quadratic form sum. That is, we put

formula here

for f∈C∞c (all integrals are on RN and are with respect to Lebesgue measure), and note that the closure of the form is non-negative and has domain equal to the Sobolev space Wm,2. We then assume that the potential has quadratic form bound less than 1 with respect to Q0, and define

formula here

This form is closed and is associated with a semibounded self-adjoint operator H in L2 (see [17, p. 348; 5, Theorem 4.23]). One can then ask whether the semigroup e−Ht defined on L2 for t[ges ]0 is extendable to a strongly continuous one-parameter semigroup on Lp for other values of p, and if so whether one can describe the domain and spectrum of its generator.

Various notions of dimension for discrete groups are compared. A group is exhibited that acts with finite stabilizers on an acyclic 2-complex in such a way that the fixed point subcomplex for any non-trivial finite subgroup is contractible, but such that the group does not admit any such action on a contractible 2-complex. This group affords a counterexample to a natural generalization of the Eilenberg–Ganea conjecture.

The paper gives a condition that is sufficient for E. Bishop's property (β), and it applies this result to the Cesàro operator on the Hardy spaces Hp, 1<p<∞. In the process, it identifies the approximate point spectrum of the Cesàro operator.

Let us consider the boundary value problem

formula here

where Ω ⊂ ℝN is a bounded domain with smooth boundary (for example, such that certain Sobolev imbedding theorems hold). Let

formula here

Then, if ϕ(s) = [mid ]s[mid ]p−1s, p > 1, problem (1) is fairly well understood and a great variety of existence results are available. These results are usually obtained using variational methods, monotone operator methods or fixed point and degree theory arguments in the Sobolev space W1,0p(Ω). If, on the other hand, we assume that ϕ is an odd nondecreasing function such that

formula here

and

formula here

then a Sobolev space setting for the problem is not appropriate and very general results are rather sparse. The first general existence results using the theory of monotone operators in Orlicz–Sobolev spaces were obtained in [5] and in [9, 10]. Other recent work that puts the problem into this framework is contained in [2] and [8].

It is in the spirit of these latter papers that we pursue the study of problem (1) and we assume that F[ratio ]Ω×ℝ→ℝ is a Carathéodory function that satisfies certain growth conditions to be specified later.

We note here that the problems to be studied, when formulated as operator equations, lead to the use of the topological degree for multivalued maps (cf. [4, 16]).

We shall see that a natural way of formulating the boundary value problem will be a variational inequality formulation of the problem in a suitable Orlicz–Sobolev space. In order to do this we shall have need of some facts about Orlicz–Sobolev spaces which we shall give now.

The definable fundamental group of a definable set in an $o$ -minimal expansion of a field is computed. This is achieved by proving the relevant case of the $o$ -minimal van Kampen theorem. This result is applied to show that if the geometrical realization of a simplicial complex over an $o$ -minimal expansion of a field is a definable manifold of dimension not 4, then its geometrical realization over the reals is a topological manifold.

Benford's law (to base $B$) for an infinite sequence $\{x_k: k \ge 1\}$ of positive quantities $x_k$ is the assertion that $\{ \log_B x_k : k \ge 1\}$ is uniformly distributed $(\bmod\ 1)$. The $3x+1$ function $T(n)$ is given by $T(n)=(3n+1)/{2}$ if $n$ is odd, and $T(n)= n/2$ if $n$ is even. This paper studies the initial iterates $x_k= T^{(k)}(x_0)$ for $1 \le k \le N$ of the $3x+1$ function, where $N$ is fixed. It shows that for most initial values $x_0$, such sequences approximately satisfy Benford's law, in the sense that the discrepancy of the finite sequence $\{\log_B x_k: 1 \le k \le N \}$ is small.

The relation between compactly generated groups of polynomial growth and FC−-groups is clarified. It follows that L1(G) is symmetric for G compactly generated and of polynomial growth.

For any pair $i,j\ge 0$ with $i+j=1$ let ${\mathbf Bad}(i,j)$ denote the set of pairs $(\alpha,\beta)\in {\bb R}^2$ for which $\max\{\|q\alpha\|^{1/i}\|q\beta\|^{1/j}\}>c/q$ for all $q\in {\bb N}$ . Here $c=c(\alpha,\beta)$ is a positive constant. If $i=0$ the set ${\mathbf Bad}(0, 1)$ is identified with ${\bb R}\times {\mathbf Bad}$ where ${\mathbf Bad}$ is the set of badly approximable numbers. That is, ${\mathbf Bad}(0, 1)$ consists of pairs $(\alpha, \beta)$ with $\alpha\in {\bb R}$ and $\beta\in {\mathbf Bad}$ If $j=0$ the roles of $\alpha$ and $\beta$ are reversed. It is proved that the set ${\mathbf Bad}(1,0)\cap {\mathbf Bad} (0,1)\cap {\mathbf Bad}(i,j)$ has Hausdorff dimension 2, that is, full dimension. The method easily generalizes to give analogous statements in higher dimensions.

It is known that, for a transcendental entire function f, the Hausdorff dimension of J(f) satisfies 1 [les ] dimJ(f) [les ] 2. For each d ∈ (1, 2), an example of a transcendental entire function f with dimJ(f) = d is given. It is then indicated how this function can be modified to produce a transcendental meromorphic function F with one pole with dimJ(F) = d. These appear to be the first examples of Julia sets with non-integer dimensions whose dimensions have been calculated exactly.

The question of extendability of the action of a cyclic group of automorphisms of a compact Riemann surface is considered. Particular attention is paid to those cases corresponding to Singerman's list of Fuchsian groups which are not finitely-maximal, and more generally to cases involving a Fuchsian triangle group. The results provide partial answers to the question of which cyclic groups are the full automorphism group of some Riemann surface of given genus g>1.

The structure of the continuous strict completely positive linear maps between locally $C^{*}$ -algebras is described.

The starting point of our investigation is the remarkable paper [2] in which Bestvina and Brady gave an example of an infinitely related group of type FP2. The result about right-angled Artin groups behind their example is best interpreted by means of the Bieri–Strebel–Neumann–Renz Σ-invariants.

For a group G the invariants Σn(G) and Σn(G, ℤ) are sets of non-trivial homomorphisms χ[ratio ]G→ℝ. They contain full information about finiteness properties of subgroups of G with abelian factor groups. The main result of [2] determines for the canonical homomorphism χ, taking each generator of the right-angled Artin group G to 1, the maximal n with χ ∈ Σn(G), respectively χ ∈ Σn(G, ℤ).

In [6] Meier, Meinert and VanWyk completed the picture by computing the full Σ-invariants of right-angled Artin groups using as well the result of Bestvina and Brady as algebraic techniques from Σ-theory. Here we offer a new account of their result which is totally geometric. In fact, we return to the Bestvina–Brady construction and simplify their argument considerably by bringing a more general notion of links into play. At the end of the first section we re-prove their main result. By re-computing the full Σ-invariants, we show in the second section that the simplification even adds some power to the method. The criterion we give provides new insight on the geometric nature of the ‘n-domination’ condition employed in [6].

The paper describes the cell structure of the affine Temperley–Lieb algebra with respect to a monomial basis. A diagram calculus is constructed for this algebra.

We show that, if $A$ is a separable simple unital $C^{*}$-algebra that absorbs the Jiang–Su algebra ${{\mathcal Z}}$ tensorially and that has real rank zero and finite decomposition rank, then $A$ is tracially approximately finite-dimensional in the sense of Lin, without any restriction on the tracial state space. As a consequence, the Elliott conjecture is true for the class of $C^{*}$-algebras as above that, additionally, satisfy the universal coefficients theorem. In particular, such algebras are completely determined by their ordered $K$-theory. They are approximately homogeneous of topological dimension less than or equal to three, approximately subhomogeneous of topological dimension at most two and their decomposition rank also is no greater than two.

Combining Le Cam's operator approach with Bergström's identity, estimates are obtained for convolutions of compound measures. The results are exemplified by consideration of asymptotic expansions for Compound Poisson approximation in Le Cam's theorem, approximation of nearly homogeneous portfolios and convergence to Compound Poisson laws.

We consider digital expansions with respect to complex integer bases. We derive precise information about the length of these expansions and the corresponding sum-of-digits function. Furthermore we give an asymptotic formula for the sum-of-digits function in large circles and prove that this function is uniformly distributed with respect to the argument. Finally the summatory function of the sum-of-digits function along the real axis is analyzed.

An example is given of a direct summand of a serial module that does not admit an indecomposable decomposition.

the set of squares $n^2$, $n<2^k$, is considered and the sum of binary digits $s(n^2)$ is split up into two parts $s_{[<k]}(n^2)+s_{[\ge k]}(n^2)$, where $s_{[<k]}(n^2) = s(n^2{\rm mod}2^k)$ collects the first $k$ digits and $s_{[\ge k]}(n^2) = s(\lfloor n^2/2^k\rfloor)$ collects the remaining digits. very precise results on the distribution of $s_{[<k]}(n^2)$ and $s_{[\ge k]}(n^2)$ are presented. for example, asymptotic formulae are provided for the numbers $\#\{n< 2^k{:} s_{[<k]}(n^2) = m\}$ and $\#\{n< 2^k {:} s_{[\ge k]}(n^2) = m\}$ and it is shown that these partial sum-of-digits functions are asymptotically equidistributed in residue classes. these results are prompted by a conjecture by gelfond saying that the (total) sum-of-digits function $s(n^2)$ is asymptotically equidistributed in residue classes.

We prove, correct and extend several results of an earlier paper of ours (using and recalling several of our later papers) about the derived functors of projective limit in abelian categories. In particular we prove that if C is an abelian category satisfying the Grothendieck axioms AB3 and AB4* and having a set of generators then the first derived functor of projective limit vanishes on so-called Mittag-Leffler sequences in C. The recent examples given by Deligne and Neeman show that the condition that the category has a set of generators is necessary. The condition AB4* is also necessary, and indeed we give for each integer $m \geq 1$ an example of a Grothendieck category Cm and a Mittag-Leffler sequence in Cm for which the derived functors of its projective limit vanish in all positive degrees except m. This leads to a systematic study of derived functors of infinite products in Grothendieck categories. Several explicit examples of the applications of these functors are also studied.

The Fourier series of the elements in the generalized Bergman spaces bp, q of harmonic functions over D and over [Copf ] (as well as those of holomorphic functions) is analysed. It is shown that the trigonometric system Ω = {r[mid ]k[mid ]eikϕ}k∈ℤ is never a basis of b1, 1 and b∞, 0 for any weighted L1-norm and L∞-norm over D. The same result holds in the special case of Bargmann–Fock space over [Copf ] (with respect to the weighted L1-norms and L∞-norms) which answers a question of Garling and Wojtaszczyk. On the other hand examples are given of weighted L1-norms and L∞-norms over [Copf ] where Ω is indeed a basis of b1, 1 and b∞, 0. Moreover, using similar methods, a weight is constructed on D where b∞, ∞ is not isomorphic to l∞ which shows that there are weighted spaces whose Banach space classifications differ completely from those which have been characterized so far.