The AF structure of non commutative toroidal Z/4Z orbifolds

For any irrational theta and rational number p/q such that q|qtheta-p|<1, a projection e of trace q|qtheta-p| is constructed in the the irrational rotation algebra A_theta that is invariant under the Fourier transform. (The latter is the order four automorphism U mapped to V, V mapped to U^{-1}, where U, V are the canonical unitaries generating A_theta.) Further, the projection e is approximately central, the cut down algebra eA_theta e contains a Fourier invariant q x q matrix algebra whose unit is e, and the cut downs eUe, eVe are approximately inside the matrix algebra. (In particular, there are Fourier invariant projections of trace k|qtheta-p| for k=1,...,q.) It is also shown that for all theta the crossed product A_theta rtimes Z_4 satisfies the Universal Coefficient Theorem. (Z_4 := Z/4Z.) As a consequence, using the Classification Theorem of G. Elliott and G. Gong for AH-algebras, a theorem of M. Rieffel, and by recent results of H. Lin, we show that A_theta rtimes Z_4 is an AF-algebra for all irrational theta in a dense G_delta.Comment: 35 page

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General Relativistic Contributions in Transformation Optics

One potentially realistic specification for devices designed with transformation optics is that they operate with high precision in curved space-time, such as Earth orbit. This raises the question of what, if any, role does space-time curvature play in determining transformation media? Transformation optics has been based on a three-vector representation of Maxwell's equations in flat Minkowski space-time. I discuss a completely covariant, manifestly four-dimensional approach that enables transformations in arbitrary space-times, and demonstrate this approach for stable circular orbits in the spherically symmetric Schwarzschild geometry. Finally, I estimate the magnitude of curvature induced contributions to satellite-borne transformation media in Earth orbit and comment on the level of precision required for metamaterial fabrication before such contributions become important.Comment: 14 pages, 3 figures. Latest version has expanded analysis, corresponds to published versio

Classical and quantum ergodicity on orbifolds

We extend to orbifolds classical results on quantum ergodicity due to Shnirelman, Colin de Verdi\`ere and Zelditch, proving that, for any positive, first-order self-adjoint elliptic pseudodifferential operator P on a compact orbifold X with positive principal symbol p, ergodicity of the Hamiltonian flow of p implies quantum ergodicity for the operator P. We also prove ergodicity of the geodesic flow on a compact Riemannian orbifold of negative sectional curvature.Comment: 14 page

'Underlying Energy Efficiency' in the US

The promotion of US energy efficiency policy is seen as a very important activity by the Energy Information Agency (EIA). Generally, the level of energy efficiency of a state is approximated by energy intensity, commonly calculated as the ratio of energy use to GDP. However, energy intensity is not an accurate proxy for energy efficiency, because changes in energy intensity are a function of changes in several factors including the structure of the economy, climate, efficiency in the use of resources and technical change. The aim of this paper is to measure the ‘underlying energy efficiency’ for the whole economy of 49 ‘states’ in the US using a stochastic frontier energy demand approach. A total US energy demand frontier function is estimated using panel data for 49 ‘states’ over the period 1995 to 2009 using several panel data models: the pooled model; the random effects model; true fixed effects model; the true random effects model; and the Mundlak versions of the pooled and random effects models. The analysis confirms that energy intensity is not a good indicator of energy efficiency; whereas, by controlling for a range of economic and other factors, the measure of ‘underlying energy efficiency’ obtained via the approach adopted here (based on the microeconomic theory of production) is

Wavelets and graph C∗C^*-algebras

Here we give an overview on the connection between wavelet theory and representation theory for graph $C^{\ast}$-algebras, including the higher-rank graph $C^*$-algebras of A. Kumjian and D. Pask. Many authors have studied different aspects of this connection over the last 20 years, and we begin this paper with a survey of the known results. We then discuss several new ways to generalize these results and obtain wavelets associated to representations of higher-rank graphs. In \cite{FGKP}, we introduced the "cubical wavelets" associated to a higher-rank graph. Here, we generalize this construction to build wavelets of arbitrary shapes. We also present a different but related construction of wavelets associated to a higher-rank graph, which we anticipate will have applications to traffic analysis on networks. Finally, we generalize the spectral graph wavelets of \cite{hammond} to higher-rank graphs, giving a third family of wavelets associated to higher-rank graphs

Glutamate dehydrogenase (GLUD1) expression in breast cancer

Dysregulated cellular metabolism is regarded as one of the hallmarks of cancer with some tumours utilising the glutamine metabolism pathway for their sustained proliferation and survival. Glutamate dehydrogenase (GLUD1) is a key enzyme in glutaminolysis converting glutamate to α-Ketoglutarate for entry into the TCA cycle. Breast cancer (BC) comprises a heterogeneous group of tumours in terms of molecular biology and clinical behaviour, and we have previously shown that altered glutamine metabolism varies substantially among the different molecular subtypes. We hypothesise that the prognostic value of GLUD1 expression will differ between the BC molecular subtypes and may act as a potential therapeutic target for BC tumours.Methods: GLUD1 was assessed at the DNA, mRNA (n=1,980) and protein (n=1,300) levels in large and well-characterised cohorts and correlated with clinicopathological parameters, molecular subtypes, patient outcome and treatments. Results: There was a correlation between GLUD1 mRNA and GLUD1 protein expression which were highly expressed in low grade Luminal/ER+ BC (

Picosecond-resolution single-photon time lens for temporal mode quantum processing

Techniques to control the spectro-temporal properties of quantum states of light at ultrafast time scales are crucial for numerous applications in quantum information science. In this work, we report an all-optical time lens for quantum signals based on Bragg-scattering four-wave mixing with picosecond resolution. Our system achieves a temporal magnification factor of 158 with single-photon level inputs, which is sufficient to overcome the intrinsic timing jitter of superconducting nanowire single-photon detectors. We demonstrate discrimination of two terahertz-bandwidth, single-photon-level pulses with 2.1 ps resolution (electronic jitter corrected resolution of 1.25 ps).We draw on elegant tools from Fourier optics to further show that the time-lens framework can be extended to perform complex unitary spectro-temporal transformations by imparting optimized temporal and spectral phase profiles to the input waveforms. Using numerical optimization techniques, we show that a four-stage transformation can realize an efficient temporal mode sorter that demultiplexes 10 Hermite–Gaussian (HG) modes. Our time-lens-based framework represents a new toolkit for arbitrary spectro-temporal processing of single photons, with applications in temporal mode quantum processing, high-dimensional quantum key distribution, temporal mode matching for quantum networks, and quantum-enhanced sensing with time-frequency entangled states.Chaitali Joshi, Ben M. Sparkes, Alessandro Farsi, Thomas Gerrits, Varun Verma, Sven Ramelow, Sae Woo Nam, and Alexander L. Gaet