Let G be a torsion-free discrete group, and let Q denote the field of algebraic numbers in C. We prove that QG fulfills the Atiyah conjecture if G lies in a certain class of groups D, which contains in particular all groups that are residually torsion-free elementary amenable or are residually free. This result implies that there are no nontrivial zero divisors in CG. The statement relies on new approximation results for L 2 -Betti numbers over QG, 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 L 2 -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 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 G.

We introduce Arbor, a performance portable library for simulation of large networks of multi-compartment neurons on HPC systems. Arbor is open source software, developed under the auspices of the HBP. The performance portability is by virtue of back-end specific optimizations for x86 multicore, Intel KNL, and NVIDIA GPUs. When coupled with low memory overheads, these optimizations make Arbor an order of magnitude faster than the most widely-used comparable simulation software. The single-node performance can be scaled out to run very large models at extreme scale with efficient weak scaling.

We study Harper operators and the closely related discrete magnetic Laplacians (DML) on a graph with a free action of a discrete group, as defined by Sunada [Sun]. A main result in this paper is that the spectral density function of DMLs associated to rational weight functions on graphs with a free action of an amenable discrete group can be approximated by the average spectral density function of the DMLs on a regular exhaustion, with either Dirichlet or Neumann boundary conditions. This then gives a criterion for the existence of gaps in the spectrum of the DML, as well as other interesting spectral properties of such DMLs. The technique used incorporates some results of algebraic number theory. © 2002 Elsevier Science (USA)

A brief overview of Forman's discrete Morse theory is presented, from which analogues of the main results of classical Morse theory can be derived for discrete Morse functions, these being functions mapping the set of cells of a CW complex to the real numbers satisfying some combinatorial relations. The discrete analogue of the strong Morse inequality was proved by Forman for finite CW complexes using a Witten deformation technique. This deformation argument is adapted to provide strong Morse inequalities for infinite CW complexes which have a finite cellular domain under the free cellular action of a discrete group. The inequalities derived are analogous to the L 2 Morse inequalities of Novikov and Shubin and the asymptotic L 2 Morse inequalities of an inexact Morse 1-form as derived by Mathai and Shubin. We also obtain quantitative lower bounds for the Morse numbers whenever the spectrum of the Laplacian contains zero, using the extended category of Farber. Academic Press

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