Absolutely continuous spectrum for a random potential on a tree with strong transverse correlations and large weighted loops

We consider random Schr\"odinger operators on tree graphs and prove absolutely continuous spectrum at small disorder for two models. The first model is the usual binary tree with certain strongly correlated random potentials. These potentials are of interest since for complete correlation they exhibit localization at all disorders. In the second model we change the tree graph by adding all possible edges to the graph inside each sphere, with weights proportional to the number of points in the sphere.Comment: 25 pages, 4 figure

Duality and KPZ in Liouville Quantum Gravity

We present a (mathematically rigorous) probabilistic and geometrical proof of the KPZ relation between scaling exponents in a Euclidean planar domain D and in Liouville quantum gravity. It uses the properly regularized quantum area measure d\mu_\gamma=\epsilon^{\gamma^2/2} e^{\gamma h_\epsilon(z)}dz, where dz is Lebesgue measure on D, \gamma is a real parameter, 0\leq \gamma <2, and h_\epsilon(z) denotes the mean value on the circle of radius \epsilon centered at z of an instance h of the Gaussian free field on D. The proof extends to the boundary geometry. The singular case \gamma >2 is shown to be related to the quantum measure d\mu_{\gamma'}, \gamma' < 2, by the fundamental duality \gamma\gamma'=4.Comment: 4 pages, 1 figur

Long-distance sound propagation over discontinuous impedances

A calculation method is presented for sound propagation over an impedance discontinuity in flat ground with a homogeneous, still atmosphere. The method is based on an approximate solution to a two dimensional boundary integral equation formulation of the problem, which expresses the wave field as the solution for homogeneous ground plus an integral over half of the boundary. Through recognizing this integral as a generalized Fourier integral, asymptotic methods are applied to evaluate the part of the integral most expensive to compute by numerical quadrature. Single frequency excess attenuation results for propagation from a point source above rigid ground to a receiver above absorbing ground are discussed. The results are applied, with air attenuation and A-weighting, to a notional jet engine noise source. Simple trends are noted

Hex Player—a virtual musical controller

In this paper, we describe a playable musical interface for tablets and multi-touch tables. The interface is a generalized keyboard, inspired by the Thummer, and consists of an array of virtual buttons. On a generalized keyboard, any given interval always has the same shape (and therefore fingering); furthermore, the fingering is consistent over a broad range of tunings. Compared to a physical generalized keyboard, a virtual version has some advantages—notably, that the spatial location of the buttons can be transformed by shears and rotations, and their colouring can be changed to reflect their musical function in different scales. We exploit these flexibilities to facilitate the playing not just of conventional Western scales but also a wide variety of microtonal generalized diatonic scales known as moment of symmetry, or well-formed, scales. A user can choose such a scale, and the buttons are automatically arranged so their spatial height corresponds to their pitch, and buttons an octave apart are always vertically above each other. Furthermore, the most numerous scale steps run along rows, while buttons within the scale are light-coloured, and those outside are dark or removed. These features can aid beginners; for example, the chosen scale might be the diatonic, in which case the piano’s familiar white and black colouring of the seven diatonic and five chromatic notes is used, but only one scale fingering need ever be learned (unlike a piano where every key needs a different fingering). Alternatively, it can assist advanced composers and musicians seeking to explore the universe of unfamiliar microtonal scales

Stereoisomerism in pentaerythritol-bridged cyclotriphosphazene tri-spiranes: spiro and ansa 1,3-propanediyldioxy disubstituted derivatives

Four isomeric products were isolated and purified from the reaction of 1,3-propanediol with the tetra-spirane cyclophosphazene-organophosphate compound (1): viz. the di-monospiro (2a), di-monoansa (2b) and two monospiro-monoansa derivatives (2c) and (2d). It is shown by 31P NMR spectroscopy on addition of a chiral solvating agent (CSA) that both the di-monospiro (2a) and di-monoansa (2b) derivatives are racemates, as expected, whereas no splitting of NMR signals occurred on addition of CSA to solutions of (2c) and (2d). It is found by X-ray crystallography that the two monospiro-monoansa spirane derivatives, (2c) and (2d), are meso diastereoisomers, which represent a new case of the stereochemistry of bis di-substituted cyclophosphazene derivatives of (1). It is also observed from the 31P NMR spectrum of the reaction mixture, supported by the yields of pure compounds, that formation of a spiro group is about 4.5 times more likely than that of an ansa moiety under the conditions of the reaction

Competitive formation of spiro and ansa derivatives in the reactions of tetrafluorobutane-1,4-diol with hexachlorocyclotriphosphazene: a comparison with butane-1,4-diol

Reaction of hexachlorocyclotriphosphazene, N3P3Cl6 (1), in two stoichiometries (1:1.2 and 1:3) with the sodium derivative of the fluorinated diol, 2,2,3,3-tetrafluorobutane-1,4-diol, (2), in THF solution at room temperature afforded six products, whose structures have been characterized by X-ray crystallography and 1H, 19F and 31P NMR spectroscopy: the mono-spiro compound, N3P3Cl4(OCH2CF2CF2CH2O), (3), its ansa isomer, (4), a di-spiro derivative N3P3Cl2(OCH2CF2CF2CH2O)2, (5), its spiro-ansa (6) and non-gem cis bis-ansa (7) isomers and a tri-spiro compound N3P3(OCH2CF2CF2CH2O)3, (8). The tri-spiro derivative (8) was also formed in the reaction of the ansa compound (4) with diol (2) in a 1:3 ratio in THF at room temperature. The reactions of (1) with step-wise additions of (2) were also investigated at low temperature (-780C) to give the same range of products as at room temperature. The results of all reactions are compared with previous work on the reactions of (1) with butane-1,4-diol/pyridine mixtures and with the reaction of hexafluorocyclotriphosphazene, N3P3F6 (9), with the silyl derivative of the diol (2), (Me3SiOCH2CF2)2, in a 1:0.4 mole ratio in the same solvent, THF

Structural investigations of phosphorus-nitrogen compounds. 7. Relationships between physical properties, electron densities, reaction mechanisms and hydrogen-bonding motifs of N3P3Cl(6-n)(NHBut)(n) derivatives

A series of compounds of the N3P3Cl(6-n)(NHBut)n family (where n = 0, 1, 2, 4 and 6) are presented and their molecular parameters are related to trends in physical properties, which provides insight into a potential reaction mechanism for nucleophilic substitution. The crystal structures of N3P3Cl5(NHBut) and N3P3Cl2(NHBut)4 have been determined at 120K and those of N3P3Cl6 and N3P3Cl4(NHBut)2 have been re-determined at 120K. These are compared with the known structure of N3P3(NHBut)6 studied at 150K. Trends in molecular parameters (phosphazene ring, P-Cl & P-N(HBut) distances, PCl2 angles and endo- and exo-cyclic phosphazene ring parameters) across the series are observed. Hydrogen-bonding motifs are identified, characterised and compared. Both the molecular and hydrogen bonding parameters are related to the electron distribution in bonds and the derived basicities of the cyclophosphazene series of compounds. These findings provide evidence for a proposed mechanism for nucleophilic substitution at a phosphorus site bearing a PCl(NHBut) moiety

The Transformation of Sediment Into Rock : Insights From IODP Site U1352, Canterbury Basin, New Zealand

ACKNOWLEDGMENTS We thank the crew of the RV JOIDES Resolution for professional seamanship, excellent drilling, and the scientific support on board. GHB and SCG thank the Australia–New Zealand IODP Consortium (ANZIC), and KMM thanks the Consortium for Ocean Leadership U.S. Science Support Program for partly funding this work. Thanks also to funding agencies of the respective authors, and Mark Lawrence (GNS Science) and Cam Nelson (University of Waikato) for their thoughtful comments on an earlier draft. Karsten Kroeger (GNS Science) helped by providing compaction data for New Zealand basins, and Michelle Kominz (Western Michigan University) provided data on which Figure 8 was developed. Further improvements were the result of thoughtful detailed reviews by Gemma Barrie, Bill Heins, Stan Paxton, Associate Editor Joe Macquaker, and Editor Leslie Melim.Peer reviewedPostprin