Lower Bounds on the Bounded Coefficient Complexity of Bilinear Maps

We prove lower bounds of order $n\log n$ for both the problem to multiply polynomials of degree $n$, and to divide polynomials with remainder, in the model of bounded coefficient arithmetic circuits over the complex numbers. These lower bounds are optimal up to order of magnitude. The proof uses a recent idea of R. Raz [Proc. 34th STOC 2002] proposed for matrix multiplication. It reduces the linear problem to multiply a random circulant matrix with a vector to the bilinear problem of cyclic convolution. We treat the arising linear problem by extending J. Morgenstern's bound [J. ACM 20, pp. 305-306, 1973] in a unitarily invariant way. This establishes a new lower bound on the bounded coefficient complexity of linear forms in terms of the singular values of the corresponding matrix. In addition, we extend these lower bounds for linear and bilinear maps to a model of circuits that allows a restricted number of unbounded scalar multiplications.Comment: 19 page

The effect of prototyping material on verbal and non-verbal behaviours in collaborative design tasks

This paper reports a study of 23 controlled experiments, with a total of 99 individual tasks, between pairs of designers collaborating to solve a simple design task using four different types of prototyping media. The aim of the study was to correlate verbal and non-verbal behaviours across different types of media with a range of measurement indicators. Using innovative movement trail images we show how collaborative sketching activity results in attenuated use of interpersonal collaborative space when compared with cardboard, clay, and Lego, which provoked intensive collaboration. Furthermore, the sketching (control) condition resulted in pre-conceived ideas being executed when compared with the three-dimensional media, where ideas emerged through collaboration. This finding suggests that increased creativity in design can result through the careful choice of prototyping media at the beginning of the design process

Parametrising arbitrary galaxy morphologies: potentials and pitfalls

We demonstrate that morphological observables (e.g. steepness of the radial light profile, ellipticity, asymmetry) are intertwined and cannot be measured independently of each other. We present strong arguments in favour of model-based parametrisation schemes, namely reliability assessment, disentanglement of morphological observables, and PSF modelling. Furthermore, we demonstrate that estimates of the concentration and Sersic index obtained from the Zurich Structure & Morphology catalogue are in excellent agreement with theoretical predictions. We also demonstrate that the incautious use of the concentration index for classification purposes can cause a severe loss of the discriminative information contained in a given data sample. Moreover, we show that, for poorly resolved galaxies, concentration index and M_20 suffer from strong discontinuities, i.e. similar morphologies are not necessarily mapped to neighbouring points in the parameter space. This limits the reliability of these parameters for classification purposes. Two-dimensional Sersic profiles accounting for centroid and ellipticity are identified as the currently most reliable parametrisation scheme in the regime of intermediate signal-to-noise ratios and resolutions, where asymmetries and substructures do not play an important role. We argue that basis functions provide good parametrisation schemes in the regimes of high signal-to-noise ratios and resolutions. Concerning Sersic profiles, we show that scale radii cannot be compared directly for profiles of different Sersic indices. Furthermore, we show that parameter spaces are typically highly nonlinear. This implies that significant caution is required when distance-based classificaton methods are used.Comment: 18 pages, 13 figure

Diverse Structural Evolution at z > 1 in Cosmologically Simulated Galaxies

From mock Hubble Space Telescope images, we quantify non-parametric statistics of galaxy morphology, thereby predicting the emergence of relationships among stellar mass, star formation, and observed rest-frame optical structure at 1 < z < 3. We measure automated diagnostics of galaxy morphology in cosmological simulations of the formation of 22 central galaxies with 9.3 < log10 M_*/M_sun < 10.7. These high-spatial-resolution zoom-in calculations enable accurate modeling of the rest-frame UV and optical morphology. Even with small numbers of galaxies, we find that structural evolution is neither universal nor monotonic: galaxy interactions can trigger either bulge or disc formation, and optically bulge-dominated galaxies at this mass may not remain so forever. Simulated galaxies with M_* > 10^10 M_sun contain relatively more disc-dominated light profiles than those with lower mass, reflecting significant disc brightening in some haloes at 1 < z < 2. By this epoch, simulated galaxies with specific star formation rates below 10^-9.7 yr^-1 are more likely than normal star-formers to have a broader mix of structural types, especially at M_* > 10^10 M_sun. We analyze a cosmological major merger at z ~ 1.5 and find that the newly proposed MID morphology diagnostics trace later merger stages while G-M20 trace earlier ones. MID is sensitive also to clumpy star-forming discs. The observability time of typical MID-enhanced events in our simulation sample is less than 100 Myr. A larger sample of cosmological assembly histories may be required to calibrate such diagnostics in the face of their sensitivity to viewing angle, segmentation algorithm, and various phenomena such as clumpy star formation and minor mergers.Comment: 23 pages, 16 figures, MNRAS accepted versio

Cerebrospinal Fluid Tau Protein Levels and F-18-Fluorodeoxyglucose Positron Emission Tomography in the Differential Diagnosis of Alzheimer's Disease

Aims: In this study, we aimed to compare cerebrospinal fluid (CSF) levels of total tau (t-tau), phosphorylated tau (p-tau(181)) and positron emission tomography with F-18-fluorodeoxyglucose (FDG-PET) in the differential diagnosis of Alzheimer's disease (AD) under clinical conditions. Method: In a cross-sectional, blinded, single-center study, we examined a sample of 75 unselected memory clinic patients with clinical diagnoses of dementia of Alzheimer type (DAT; n = 24), amnestic mild cognitive impairment (MCI; n = 16), other dementias (n = 13) and nondemented controls (n = 22). Discriminative accuracy, sensitivity and specificity were calculated and compared using ROC analyses. Results: p-tau(181) and FDG-PET were comparable in separating DAT from controls (sensitivity: 67 vs. 79%; specificity: 91% for both) and patients with other dementias (sensitivity: 71 vs. 79%; specificity: 100% for both). The sensitivity of p-tau 181 in differentiating MCI patients from controls was significantly (p < 0.05) superior to that of FDG-PET (75 vs. 44%) at a comparably high specificity (82 vs. 91%); t-tau measures were less accurate in all analyses. Conclusions: FDG-PET and CSF p-tau(181) levels are able to discriminate DAT in heterogeneous and unselected samples with a high accuracy. CSF p-tau(181) might be somewhat superior for a sensitive detection of patients with MCI. Copyright (C) 2010 S. Karger AG, Base

Quantifying galaxy shapes: Sersiclets and beyond

Parametrising galaxy morphologies is a challenging task, e.g., in shear measurements of weak lensing or investigations of galaxy evolution. The huge variety of morphologies requires an approach that is highly flexible, e.g., accounting for azimuthal structure. We revisit the method of sersiclets, where galaxy morphologies are decomposed into basis functions based on the Sersic profile. This approach is justified by the fact that the Sersic profile is the first-order Taylor expansion of any real light profile. We show that sersiclets overcome the modelling failures of shapelets. However, sersiclets implicate an unphysical relation between the steepness of the light profile and the spatial scale of azimuthal structures, which is not obeyed by real galaxy morphologies and can therefore give rise to modelling failures. Moreover, we demonstrate that sersiclets are prone to undersampling, which restricts sersiclet modelling to highly resolved galaxy images. Analysing data from the Great08 challenge, we demonstrate that sersiclets should not be used in weak-lensing studies. We conclude that although the sersiclet approach appears very promising at first glance, it suffers from conceptual and practical problems that severly limit its usefulness. The Sersic profile can be enhanced by higher-order terms in the Taylor expansion, which can drastically improve model reconstructions of galaxy images. If orthonormalised, these higher-order profiles can overcome the problems of sersiclets while preserving their mathematical justification.Comment: 14 pages, 12 figures, 2 tables; accepted by MNRA

Beyond Spheroids and Discs: Classifications of CANDELS Galaxy Structure at 1.4 < z < 2 via Principal Component Analysis

Important but rare and subtle processes driving galaxy morphology and star-formation may be missed by traditional spiral, elliptical, irregular or S\'ersic bulge/disk classifications. To overcome this limitation, we use a principal component analysis of non-parametric morphological indicators (concentration, asymmetry, Gini coefficient, $M_{20}$, multi-mode, intensity and deviation) measured at rest-frame $B$-band (corresponding to HST/WFC3 F125W at 1.4 $10^{10} M_{\odot}$) galaxy morphologies. Principal component analysis (PCA) quantifies the correlations between these morphological indicators and determines the relative importance of each. The first three principal components (PCs) capture $\sim$75 per cent of the variance inherent to our sample. We interpret the first principal component (PC) as bulge strength, the second PC as dominated by concentration and the third PC as dominated by asymmetry. Both PC1 and PC2 correlate with the visual appearance of a central bulge and predict galaxy quiescence. PC1 is a better predictor of quenching than stellar mass, as as good as other structural indicators (S\'ersic-n or compactness). We divide the PCA results into groups using an agglomerative hierarchical clustering method. Unlike S\'ersic, this classification scheme separates compact galaxies from larger, smooth proto-elliptical systems, and star-forming disk-dominated clumpy galaxies from star-forming bulge-dominated asymmetric galaxies. Distinguishing between these galaxy structural types in a quantitative manner is an important step towards understanding the connections between morphology, galaxy assembly and star-formation.Comment: 31 pages, 24 figures, accepted for publication in MNRA

Constraints on the Assembly and Dynamics of Galaxies: I. Detailed Rest-frame Optical Morphologies on Kiloparsec-scale of z ~ 2 Star-forming Galaxies

We present deep and high-resolution HST/NIC2 F160W imaging at 1.6micron of six z~2 star-forming galaxies with existing near-IR integral field spectroscopy from SINFONI at the VLT. The unique combination of rest-frame optical imaging and nebular emission-line maps provides simultaneous insight into morphologies and dynamical properties. The overall rest-frame optical emission of the galaxies is characterized by shallow profiles in general (Sersic index n<1), with median effective radii of ~5kpc. The morphologies are significantly clumpy and irregular, which we quantify through a non-parametric morphological approach, estimating the Gini (G), Multiplicity (Psi), and M_20 coefficients. The strength of the rest-frame optical emission lines in the F160W bandpass indicates that the observed structure is not dominated by the morphology of line-emitting gas, and must reflect the underlying stellar mass distribution of the galaxies. The sizes and structural parameters in the rest-frame optical continuum and Halpha emission reveal no significant differences, suggesting similar global distributions of the on-going star formation and more evolved stellar population. While no strong correlations are observed between stellar population parameters and morphology within the NIC2/SINFONI sample itself, a consideration of the sample in the context of a broader range of z~2 galaxy types indicates that these galaxies probe the high specific star formation rate and low stellar mass surface density part of the massive z~2 galaxy population, with correspondingly large effective radii, low Sersic indices, low G, and high Psi and M_20. The combined NIC2 and SINFONI dataset yields insights of unprecedented detail into the nature of mass accretion at high redshift. [Abridged]Comment: 44 pages, 19 figures. Revised version accepted for publication in the Astrophysical Journa