Tensor decomposition and homotopy continuation

A computationally challenging classical elimination theory problem is to compute polynomials which vanish on the set of tensors of a given rank. By moving away from computing polynomials via elimination theory to computing pseudowitness sets via numerical elimination theory, we develop computational methods for computing ranks and border ranks of tensors along with decompositions. More generally, we present our approach using joins of any collection of irreducible and nondegenerate projective varieties $X_1,\ldots,X_k\subset\mathbb{P}^N$ defined over $\mathbb{C}$. After computing ranks over $\mathbb{C}$, we also explore computing real ranks. Various examples are included to demonstrate this numerical algebraic geometric approach.Comment: We have added two examples: A Coppersmith-Winograd tensor, Matrix multiplication with zeros. (26 pages, 1 figure

Test of Universality in the Ising Spin Glass Using High Temperature Graph Expansion

We calculate high-temperature graph expansions for the Ising spin glass model with 4 symmetric random distribution functions for its nearest neighbor interaction constants J_{ij}. Series for the Edwards-Anderson susceptibility \chi_EA are obtained to order 13 in the expansion variable (J/(k_B T))^2 for the general d-dimensional hyper-cubic lattice, where the parameter J determines the width of the distributions. We explain in detail how the expansions are calculated. The analysis, using the Dlog-Pad\'e approximation and the techniques known as M1 and M2, leads to estimates for the critical threshold (J/(k_B T_c))^2 and for the critical exponent \gamma in dimensions 4, 5, 7 and 8 for all the distribution functions. In each dimension the values for \gamma agree, within their uncertainty margins, with a common value for the different distributions, thus confirming universality.Comment: 13 figure

Early-type galaxies in the SDSS. II. Correlations between observables

A magnitude limited sample of nearly 9000 early-type galaxies, in the redshift range 0.01 < z < 0.3, was selected from the Sloan Digital Sky Survey using morphological and spectral criteria. The sample was used to study how early-type galaxy observables, including luminosity L, effective radius R_o, surface brightness I_o, color, and velocity dispersion sigma, are correlated with one another. Measurement biases are understood with mock catalogs which reproduce all of the observed scaling relations and their dependences on fitting technique. At any given redshift, the intrinsic distribution of luminosities, sizes and velocity dispersions in our sample are all approximately Gaussian. A maximum likelihood analysis shows that sigma ~ L^{0.25\pm 0.012}, R_o ~ L^{0.63\pm 0.025}, and R_o ~ I^{-0.75\pm 0.02} in the r* band. In addition, the mass-to-light ratio within the effective radius scales as M_o/L ~ L^{0.14\pm 0.02} or M_o/L ~ M_o^{0.22\pm 0.05}, and galaxies with larger effective masses have smaller effective densities: Delta_o ~ M_o^{-0.52\pm 0.03}. These relations are approximately the same in the g*, i* and z* bands. Relative to the population at the median redshift in the sample, galaxies at lower and higher redshifts have evolved only little, with more evolution in the bluer bands. The luminosity function is consistent with weak passive luminosity evolution and a formation time of about 9 Gyrs ago.Comment: 29 pages, 11 figures. Accepted by AJ (scheduled for April 2003). This paper is part II of a revised version of astro-ph/011034

Kelvin probe characterization of buried graphitic microchannels in single-crystal diamond

In this work, we present an investigation by Kelvin Probe Microscopy (KPM) of buried graphitic microchannels fabricated in single-crystal diamond by direct MeV ion microbeam writing. Metal deposition of variable-thickness masks was adopted to implant channels with emerging endpoints and high temperature annealing was performed in order to induce the graphitization of the highly-damaged buried region. When an electrical current was flowing through the biased buried channel, the structure was clearly evidenced by KPM maps of the electrical potential of the surface region overlying the channel at increasing distances from the grounded electrode. The KPM profiling shows regions of opposite contrast located at different distances from the endpoints of the channel. This effect is attributed to the different electrical conduction properties of the surface and of the buried graphitic layer. The model adopted to interpret these KPM maps and profiles proved to be suitable for the electronic characterization of buried conductive channels, providing a non-invasive method to measure the local resistivity with a micrometer resolution. The results demonstrate the potential of the technique as a powerful diagnostic tool to monitor the functionality of all-carbon graphite/diamond devices to be fabricated by MeV ion beam lithography.Comment: 21 pages, 5 figure

Polarized Diffuse Emission at 2.3 GHz in a High Galactic Latitude Area

Polarized diffuse emission observations at 2.3 GHz in a high Galactic latitude area are presented. The 2\degr X 2\degr field, centred in (\alpha=5^h,\delta=-49\degr), is located in the region observed by the BOOMERanG experiment. Our observations has been carried out with the Parkes Radio telescope and represent the highest frequency detection done to date in low emission areas. Because of a weaker Faraday rotation action, the high frequency allows an estimate of the Galactic synchrotron contamination of the Cosmic Microwave Background Polarization (CMBP) that is more reliable than that done at 1.4 GHz. We find that the angular power spectra of the E- and B-modes have slopes of \beta_E = -1.46 +/- 0.14 and \beta_B = -1.87 +/- 0.22, indicating a flattening with respect to 1.4 GHz. Extrapolated up to 32 GHz, the E-mode spectrum is about 3 orders of magnitude lower than that of the CMBP, allowing a clean detection even at this frequency. The best improvement concerns the B-mode, for which our single-dish observations provide the first estimate of the contamination on angular scales close to the CMBP peak (about 2 degrees). We find that the CMBP B-mode should be stronger than synchrotron contamination at 90 GHz for models with T/S > 0.01. This low level could move down to 60-70 GHz the optimal window for CMBP measures.Comment: 5 pages, 6 figures, accepted for publication in MNRAS Letter