CP decomposition for tensors via alternating least squares with QR decomposition

The CP tensor decomposition is used in applications such as machine learning and signal processing to discover latent low-rank structure in multidimensional data. Computing a CP decomposition via an alternating least squares (ALS) method reduces the problem to several linear least squares problems. The standard way to solve these linear least squares subproblems is to use the normal equations, which inherit special tensor structure that can be exploited for computational efficiency. However, the normal equations are sensitive to numerical ill-conditioning, which can compromise the results of the decomposition. In this paper, we develop versions of the CP-ALS algorithm using the QR decomposition and the singular value decomposition, which are more numerically stable than the normal equations, to solve the linear least squares problems. Our algorithms utilize the tensor structure of the CP-ALS subproblems efficiently, have the same complexity as the standard CP-ALS algorithm when the input is dense and the rank is small, and are shown via examples to produce more stable results when ill-conditioning is present. Our MATLAB implementation achieves the same running time as the standard algorithm for small ranks, and we show that the new methods can obtain lower approximation error.     

Crystal structure, stoichiometry, and dielectric relaxation in Bi3.32Nb7.09O22.7 and structurally related ternary phases

The crystal structure of the phase previously reported to occur at 4:9 Bi₂O₃:Nb₂O₅ has been determined using single-crystal X-ray and powder neutron diffraction (P6₃/mmc; a=7.4363(1) Å, c=19.7587(5) Å; Z=2). The structural study combined with phase equilibrium analyses indicate that the actual composition is Bi_3.32Nb_7.09O_22.7. This binary compound is the end-member of a family of four phases which form along a line between it and the pyrochlore phase field in the Bi₂O₃:Fe₂O₃:Nb₂O₅ system. The structures are derived from the parent pyrochlore end-member by chemical twinning, and can also be described as unit-cell intergrowths of the pyrochlore and hexagonal tungsten bronze (HTB) structures. The dielectric properties of the three chemically twinned pyrochlore phases, Bi_3.32Nb_7.09O_22.7, Bi_9.3Fe_1.1Nb_16.9O_57.8 and Bi_5.67FeNb₁₀O₃₅, were characterized. All exhibit low-temperature, broad dielectric relaxation similar to that of the Bi-Fe-Nb-O pyrochlore. At 1 MHz and ≈175 K the observed relative permittivites were 345, 240, and 205, respectively, compared to 125 for the Bi-Fe-Nb-O pyrochlore. The higher relative permittivities observed for the chemically twinned pyrochlore derivatives are ascribed to the presence of HTB blocks in their structures: The Bi atoms located in the HTB blocks feature highly asymmetric coordination environments compared to pyrochlore, and the magnitude of the relative permittivity increases with the proportion of Bi located within the HTB portions of the structures.     

Theory and Practice: Bulk Synthesis of C3B and its H2‐ and Li‐Storage Capacity

Previous theoretical studies of C₃B have suggested that boron‐doped graphite is a promising H₂‐ and Li‐storage material, with large maximum capacities. These characteristics could lead to exciting applications as a lightweight H₂‐storage material for automotive engines and as an anode in a new generation of batteries. However, for these applications to be realized a synthetic route to bulk C₃B must be developed. Here we show the thermolysis of a single‐source precursor (1,3‐(BBr₂)₂C₆H₄) to produce graphitic C₃B, thus allowing the characteristics of this elusive material to be tested for the first time. C₃B was found to be compositionally uniform but turbostratically disordered. Contrary to theoretical expectations, the H₂‐ and Li‐storage capacities are lower than anticipated, results that can partially be explained by the disordered nature of the material. This work suggests that to model the properties of graphitic materials more realistically, the possibility of disorder must be considered.     

Carbon stable isotopes reveal complex trophic interactions in lake plankton

The lower trophic linkages in lake plankton food webs are generally described as relatively simple, even accounting for the additional complexity of potential 'microbial looping'. Crustacean zooplankton are frequently amalgamated into one trophic functional group as grazers of autotrophic production. The carbon stable isotope ratios for separated zooplankton species, particulate organic matter (POM) and phytoplankton from a number of lakes in Finland and the UK were analysed. These revealed greater complexity in trophic interactions than would otherwise be observed if the zooplankton had been represented by a mixed sample. Grazing zooplankton were usually depleted in (13)C relative to the bulk POM on which they might feed, with (13)C deviating by up to 17 per thousand There were no consistent differences between (13)C values for copepods and cladocerans. Predatory cladocerans were generally enriched by greater than 1 per thousand compared to their putative prey. We suggest that care in separating the zooplankton species for stable isotope analysis may expose otherwise undetected sources of carbon and facilitate unravelling trophic links further up the food web. Copyright 1999 John Wiley & Sons, Ltd.     

Mid-infrared spectroscopic studies and lasing in GaAs–AlGaAs quantum cascade devices

Complementary intersubband and interband optical measurements have been employed in order to study GaAs–AlGaAs quantum cascade light-emitting diode and laser structures. Using these techniques, we have measured the redistribution of electrons throughout the bridging regions and upper states in the active regions of the diode device with increasing bias. The high quality of the sample gives very narrow line widths in the optical spectra, permitting the resolution of transitions involving closely spaced energy levels. This has allowed the direct observation of level alignment at the onset of current flow through the device. In addition, stimulated emission at λ=9.7 μm has been observed from a GaAs–AlGaAs laser structure under pulsed operation. A threshold current density of 6.5 kA/cm² and peak power 300 mW are measured at 10 K and lasing operation is observed up to 200 K.     

Combined AFM and laser lithography on hydrogen-passivated amorphous silicon

We report a novel combination of AFM lithography and laser direct writing on hydrogen-passivated amorphous silicon surfaces to fabricate combined silicon milli-, micro- and nanostructures. Selective oxidation is performed by focusing a laser beam (λ=458 nm) on a hydrogen-terminated silicon surface, forming the millimetre-size contact pads for connection of nanometre-scale patterns. The nanostructures are made by electric-field-enhanced oxidation using a contact mode AFM equipped with a metal-coated tip. Both techniques are based on selective oxidation of hydrogen-passivated amorphous silicon, where the oxide is used as an etch mask in a single etch step. The lithographic process has also been demonstrated using a reflection mode scanning near-field optical microscope with an uncoated fiber probe.     

The geometry of rank decompositions of matrix multiplication II: 3 × 3 matrices

This is the second in a series of papers on rank decompositions of the matrix multiplication tensor. We present new rank 23 decompositions for the $3\times 3$ matrix multiplication tensor $M_{〈3〉}$. All our decompositions have symmetry groups that include the standard cyclic permutation of factors but otherwise exhibit a range of behavior. One of them has 11 cubes as summands and admits an unexpected symmetry group of order 12.We establish basic information regarding symmetry groups of decompositions and outline two approaches for finding new rank decompositions of $M_{〈\mathbf{n}〉}$ for larger n.     

Crystal Chemistry of Hydrous Aluminium Arsenate Minerals

Crystallography Reports - The hydrous aluminium arsenate minerals form a small but diverse group of crystal structures. They range from mansfieldite, AlAsO4 ? 2H2O, with isolated Al-centred...     

Observation of the missing mode effect in a poly-phenylenevinylene derivative: effect of solvent, chain packing, and composition

Optical emission spectra of poly[2-methoxy-5-[3('),7(')-dimethyloctyloxy)-1,4-phenylenevinylene] (MDMO-PPV) in dilute solutions exhibit a vibronic progression interval (～1225?cm(-1)) that does not correspond to any ground state vibrational mode frequency. This phenomenon is assigned as the missing mode effect (MIME) in which five key displaced polymer backbone vibrational modes in the range of 800-1600?cm(-1) contribute to the MIME interval. Emission spectra are calculated by analytically solving the time-dependent Schro?dinger equation using estimates of mode-specific vibrational displacements determined independently from preresonance Raman intensities. Emission spectra of MDMO-PPV thin films and nanoparticles are measured and lineshapes show an increase of the MIME frequency to ～1340?cm(-1) in addition to changes in vibronic intensity distributions and energies. Composite blend thin films consisting of MDMO-PPV and a fullerene derivative (1:1 w/w) exhibit a substantially larger MIME interval (～1450?cm(-1)) that arises from an increase in polymer chain planarity. This structural change is most apparent from large decreases of the excited state displacement of an out-of-plane C-H bending mode (961?cm(-1)) that becomes forbidden in the planar structure.