Approximating L2-invariants, and the Atiyah conjecture

Let G be a torsion free discrete group and let \bar{Q} denote the field of algebraic numbers in C. We prove that \bar{Q}[G] fulfills the Atiyah conjecture if G lies in a certain class of groups D, which contains in particular all groups which are residually torsion free elementary amenable or which are residually free. This result implies that there are no non-trivial zero-divisors in C[G]. The statement relies on new approximation results for L2-Betti numbers over \bar{Q}[G], which are the core of the work done in this paper. Another set of results in the paper is concerned with certain number theoretic properties of eigenvalues for the combinatorial Laplacian on L2-cochains on any normal covering space of a finite CW complex. We establish the absence of eigenvalues that are transcendental numbers, whenever the covering transformation group is either amenable or in the Linnell class \mathcal{C}. We also establish the absence of eigenvalues that are Liouville transcendental numbers whenever the covering transformation group is either residually finite or more generally in a certain large bootstrap class \mathcal{G}. Please take the errata to Schick: "L2-determinant class and approximation of L2-Betti numbers" into account, which are added at the end of the file, rectifying some unproved statements about "amenable extension". As a consequence, throughout, amenable extensions should be extensions with normal subgroups.Comment: AMS-LaTeX2e, 33 pages; improved presentation, new and detailed proof about absence of trancendental eigenvalues; v3: added errata to "L2-determinant class and approximation of L2-Betti numbers", requires to restrict to slightly weaker statement

Arithmetic properties of eigenvalues of generalized Harper operators on graphs

Let \Qbar denote the field of complex algebraic numbers. A discrete group $G$ is said to have the $\sigma$-multiplier algebraic eigenvalue property, if for every matrix $A$ with entries in the twisted group ring over the complex algebraic numbers M_d(\Qbar(G,\sigma)), regarded as an operator on $l^2(G)^d$, the eigenvalues of $A$ are algebraic numbers, where $\sigma$ is an algebraic multiplier. Such operators include the Harper operator and the discrete magnetic Laplacian that occur in solid state physics. We prove that any finitely generated amenable, free or surface group has this property for any algebraic multiplier $\sigma$. In the special case when $\sigma$ is rational ($\sigma^n$=1 for some positive integer $n$) this property holds for a larger class of groups, containing free groups and amenable groups, and closed under taking directed unions and extensions with amenable quotients. Included in the paper are proofs of other spectral properties of such operators.Comment: 28 pages, latex2e, paper revise

Asymptotic consistency under large entropy sampling designs with unequal probabilities

A large part of survey sampling literature is devoted to unequal probabilities sampling designs without replacement. Brewer and Hanif (1983) provided a summary of these sampling designs. The maximum entropy designs is one of them. Consistency results have been proven for the maximum entropy sampling (Hájek, 1964). The aim is to give sufficient conditions under which Hájek (1964) consistency results still hold for large entropy sampling designs which are different from the maximum entropy design. These conditions involve modes of convergence of sampling designs towards the maximum entropy design. We show that these conditions are satisfied for the popular Rao-Sampford (Rao, 1965, Sampford, 1967) design. Our consistency results are applied to the Hájek (1964) simple variance estimator. This estimator does not require joint-inclusion probabilities and can be easily estimated using weighted least squares regression (Berger, 2004, 2005b). Deville (1999) conjectured that this estimator is suitable for any sampling designs (see also Brewer and Donadio, 2003). Our consistency result gives regularity conditions under which this estimator is consistent which justifies Deville’s (1999) conjecture

Age of Information in Multicast Networks with Multiple Update Streams

We consider the age of information in a multicast network where there is a single source node that sends time-sensitive updates to $n$ receiver nodes. Each status update is one of two kinds: type I or type II. To study the age of information experienced by the receiver nodes for both types of updates, we consider two cases: update streams are generated by the source node at-will and update streams arrive exogenously to the source node. We show that using an earliest $k_1$ and $k_2$ transmission scheme for type I and type II updates, respectively, the age of information of both update streams at the receiver nodes can be made a constant independent of $n$. In particular, the source node transmits each type I update packet to the earliest $k_1$ and each type II update packet to the earliest $k_2$ of $n$ receiver nodes. We determine the optimum $k_1$ and $k_2$ stopping thresholds for arbitrary shifted exponential link delays to individually and jointly minimize the average age of both update streams and characterize the pareto optimal curve for the two ages