Scales for co-compact embeddings of virtually free groups

Let $\Gamma$ be a group which is virtually free of rank at least 2 and let $\mathcal{F}_{td}(\Gamma)$ be the family of totally disconnected, locally compact groups containing $\Gamma$ as a co-compact lattice. We prove that the values of the scale function with respect to groups in $\mathcal{F}_{td}(\Gamma)$ evaluated on the subset $\Gamma$ have only finitely many prime divisors. This can be thought of as a uniform property of the family $\mathcal{F}_{td}(\Gamma)$.Comment: 12 pages; key words: uniform lattice, virtually free group, totally disconnected group, scale function (Error in references corrected in version 2

Resonant and inelastic Andreev tunneling observed on a carbon nanotube quantum dot

We report the observation of two fundamental sub-gap transport processes through a quantum dot (QD) with a superconducting contact. The device consists of a carbon nanotube contacted by a Nb superconducting and a normal metal contact. First, we find a single resonance with position, shape and amplitude consistent with the theoretically predicted resonant Andreev tunneling (AT) through a single QD level. Second, we observe a series of discrete replicas of resonant AT at a separation of $\sim145\,\mu$eV, with a gate, bias and temperature dependence characteristic for boson-assisted, inelastic AT, in which energy is exchanged between a bosonic bath and the electrons. The magnetic field dependence of the replica's amplitudes and energies suggest that two different bosons couple to the tunnel process.Comment: 5 pages + 9 pages supplementary materia

Flat rank of automorphism groups of buildings

The flat rank of a totally disconnected locally compact group G, denoted flat-rk(G), is an invariant of the topological group structure of G. It is defined thanks to a natural distance on the space of compact open subgroups of G. For a topological Kac-Moody group G with Weyl group W, we derive the inequalities: alg-rk(W)\le flat-rk(G)\le rk(|W|\_0). Here, alg-rk(W) is the maximal $\mathbb{Z}$-rank of abelian subgroups of W, and rk(|W|\_0) is the maximal dimension of isometrically embedded flats in the CAT0-realization |W|\_0. We can prove these inequalities under weaker assumptions. We also show that for any integer n \geq 1 there is a topologically simple, compactly generated, locally compact, totally disconnected group G, with flat-rk(G)=n and which is not linear

Scale-multiplicative semigroups and geometry: automorphism groups of trees

A scale-multiplicative semigroup in a totally disconnected, locally compact group $G$ is one for which the restriction of the scale function on $G$ is multiplicative. The maximal scale-multiplicative semigroups in groups acting 2-transitively on the set of ends of trees without leaves are determined in this paper and shown to correspond to geometric features of the tree.Comment: submitted to Groups, Geometry, and Dynamic

Andreev bound states probed in three-terminal quantum dots

We demonstrate several new electron transport phenomena mediated by Andreev bound states (ABSs) that form on three-terminal carbon nanotube (CNT) QDs, with one superconducting (S) contact in the center and two adjacent normal metal (N) contacts. Three-terminal spectroscopy allows us to identify the coupling to the N contacts as the origin of the Andreev resonance (AR) linewidths and to determine the critical coupling strengths to S, for which a ground state transition S-QD systems can occur. We ascribe replicas of the lowest-energy ABS resonance to transitions between the ABS and odd-parity excited QD states, a process called excited state ABS resonances. In the conductance between the two N contacts we find a characteristic pattern of positive and negative differential subgap conductance, which we explain by considering two nonlocal processes, the creation of Cooper pairs in S by electrons from both N terminals, and a novel mechanism called resonant ABS tunneling. In the latter, electrons are transferred via the ABS without creating Cooper pairs in S. The three-terminal geometry also allows spectroscopy experiments with different boundary conditions, for example by leaving S floating. Surprisingly, we find that, depending on the boundary conditions, the experiments either show single-particle Coulomb blockade resonances, ABS characteristics, or both in the same measurements, seemingly contradicting the notion of ABSs replacing the single particle states as eigenstates of the QD. We qualitatively explain these results as originating from the finite time scale required for the coherent oscillations between the superposition states after a single electron tunneling event. These experiments demonstrate that three-terminal experiments on a single complex quantum object can also be useful to investigate charge dynamics otherwise not accessible due to the very high frequencies.Comment: 15 pages, 16 figure

Fork stamping of pristine carbon nanotubes onto ferromagnetic contacts for spin-valve devices

We present a fabrication scheme called 'fork stamping' optimized for the dry transfer of individual pristine carbon nanotubes (CNTs) onto ferromagnetic contact electrodes fabricated by standard lithography. We demonstrate the detailed recipes for a residue-free device fabrication and in-situ current annealing on suspended CNT spin-valve devices with ferromagnetic Permalloy (Py) contacts and report preliminary transport characterization and magnetoresistance experiments at cryogenic temperatures. This scheme can directly be used to implement more complex device structures, including multiple gates or superconducting contacts.Comment: 7 pages, 4 figures, submitted to IWEPNM 2015 conference proceedings (physica status solidi (b)

Scavenger 0.1: A Theorem Prover Based on Conflict Resolution

This paper introduces Scavenger, the first theorem prover for pure first-order logic without equality based on the new conflict resolution calculus. Conflict resolution has a restricted resolution inference rule that resembles (a first-order generalization of) unit propagation as well as a rule for assuming decision literals and a rule for deriving new clauses by (a first-order generalization of) conflict-driven clause learning.Comment: Published at CADE 201

Contraction groups and scales of automorphisms of totally disconnected locally compact groups

We study contraction groups for automorphisms of totally disconnected locally compcat groups using the scale of the automorphism as a tool. The contraction group is shown to be unbounded when the inverse automorphism has non-trivial scale and this scale is shown to be the inverse value of the modular function on the closure of the contraction group at the automorphism. The closure of the contraction group is represented as acting on a homogenous tree and closed contraction groups are characterised.Comment: revised version, 29 pages, to appear in Israel Journal of Mathematics, please note that document starts on page

Permalloy-based carbon nanotube spin-valve

In this Letter we demonstrate that Permalloy (Py), a widely used Ni/Fe alloy, forms contacts to carbon nanotubes (CNTs) that meet the requirements for the injection and detection of spin-polarized currents in carbon-based spintronic devices. We establish the material quality and magnetization properties of Py strips in the shape of suitable electrical contacts and find a sharp magnetization switching tunable by geometry in the anisotropic magnetoresistance (AMR) of a single strip at cryogenic temperatures. In addition, we show that Py contacts couple strongly to CNTs, comparable to Pd contacts, thereby forming CNT quantum dots at low temperatures. These results form the basis for a Py-based CNT spin-valve exhibiting very sharp resistance switchings in the tunneling magnetoresistance, which directly correspond to the magnetization reversals in the individual contacts observed in AMR experiments.Comment: 3 page

Architectural Intervention: The Resilient Community Center

In the city of Centralia, Illinois, several issues govern the city’s future, all stemming from the same major problem: the economy. To intervene, this community center proposal aims to encourage economic development and growth in the city, assisting small businesses with starting and expanding in the region