Symmetrization in Geometry

The concept of an $i$-symmetrization is introduced, which provides a convenient framework for most of the familiar symmetrization processes on convex sets. Various properties of $i$-symmetrizations are introduced and the relations between them investigated. New expressions are provided for the Steiner and Minkowski symmetrals of a compact convex set which exhibit a dual relationship between them. Characterizations of Steiner, Minkowski and central symmetrization, in terms of natural properties that they enjoy, are given and examples are provided to show that none of the assumptions made can be dropped or significantly weakened. Other familiar symmetrizations, such as Schwarz symmetrization, are discussed and several new ones introduced.Comment: A chacterization of central symmetrization has been added and several typos have been corrected. This version has been accepted for publication on Advances in Mathematic

Counts and Sizes of Galaxies in the Hubble Deep Field - South: Implications for the Next Generation Space Telescope

Science objectives for the Next Generation Space Telescope (NGST) include a large component of galaxy surveys, both imaging and spectroscopy. The Hubble Deep Field datasets include the deepest observations ever made in the ultraviolet, optical and near infrared, reaching depths comparable to that expected for NGST spectroscopy. We present the source counts, galaxy sizes and isophotal filling factors of the HDF-South images. The observed integrated galaxy counts reach >500 galaxies per square arcminute at AB<30. We extend these counts to faint levels in the infrared using models. The trend previously seen that fainter galaxies are smaller, continues to AB=29 in the high resolution HDF-S STIS image, where galaxies have a typical half-light radius of 0.1 arcseconds. Extensive Monte Carlo simulations show that the small measured sizes are not due to selection effects until >29mag. Using the HDF-S NICMOS image, we show that galaxies are smaller in the near infrared than they are in the optical. We analyze the isophotal filling factor of the HDF-S STIS image, and show that this image is mostly empty sky even at the limits of galaxy detection, a conclusion we expect to hold true for NGST spectroscopy. At the surface brightness limits expected for NGST imaging, however, about a quarter of the sky is occupied by the outer isophotes of AB<30 galaxies. We discuss the implications of these data on several design concepts of the NGST near-infrared spectrograph. We compare the effects of resolution and the confusion limit of various designs, as well as the multiplexing advantages of either multi-object or full-field spectroscopy. We argue that the optimal choice for NGST spectroscopy of high redshift galaxies is a multi-object spectrograph (MOS) with target selection by a micro electro mechanical system (MEMS) device.Comment: 27 pages including 10 figures, accepted for publication in the Astronomical Journal, June 2000, abridged abstrac

On the reconstruction of planar lattice-convex sets from the covariogram

A finite subset $K$ of $\mathbb{Z}^d$ is said to be lattice-convex if $K$ is the intersection of $\mathbb{Z}^d$ with a convex set. The covariogram $g_K$ of $K\subseteq \mathbb{Z}^d$ is the function associating to each u \in \integer^d the cardinality of $K\cap (K+u)$. Daurat, G\'erard, and Nivat and independently Gardner, Gronchi, and Zong raised the problem on the reconstruction of lattice-convex sets $K$ from $g_K$. We provide a partial positive answer to this problem by showing that for $d=2$ and under mild extra assumptions, $g_K$ determines $K$ up to translations and reflections. As a complement to the theorem on reconstruction we also extend the known counterexamples (i.e., planar lattice-convex sets which are not reconstructible, up to translations and reflections) to an infinite family of counterexamples.Comment: accepted in Discrete and Computational Geometr

Expressiveness and complexity of graph logic

We investigate the complexity and expressive power of the spatial logic for querying graphs introduced by Cardelli, Gardner and Ghelli (ICALP 2002).We show that the model-checking complexity of versions of this logic with and without recursion is PSPACE-complete. In terms of expressive power, the version without recursion is a fragment of the monadic second-order logic of graphs and we show that it can express complete problems at every level of the polynomial hierarchy. We also show that it can define all regular languages, when interpretation is restricted to strings. The expressive power of the logic with recursion is much greater as it can express properties that are PSPACE-complete and therefore unlikely to be definable in second-order logic

Local reasoning about mashups

Web mashups are complex programs that dynamically compose XML data and JavaScript code from many sources. Whereas data is sometimes formally specified by XML schema, code never is. This makes it difficult to construct reliable software. Using local Hoare reasoning, introduced in separation logic to reason about e.g. C programs and extended in context logic to reason about e.g. the DOM library, we are able to reason about mashup programs, proving that they are fault-free and providing specifications for code that are analogous to XML schema for data

Absence of anomalous negative lattice-expansion for polycrystalline sample of Tb2Ti2O7

High resolution X-ray powder-diffraction experiments on a well-characterized polycrystalline sample of the spin liquid Tb2Ti2O7 reveal that it shows normal positive thermal-expansion above 4 K, which does not agree with the intriguing anomalous negative thermal-expansion due to a magneto-elastic coupling reported for a single crystal sample below 20 K. We also performed a Rietveld profile refinement of a powder-diffraction pattern taken at a room temperature, and confirmed that it is consistent with the fully ordered cubic pyrochlore structure.Comment: 2 pages, 3 figure

Galaxy Morphology from NICMOS Parallel Imaging

We present high resolution NICMOS images of random fields obtained in parallel to other HST observations. We present galaxy number counts reaching H=24. The H-band galaxy counts show good agreement with the deepest I- and K-band counts obtained from ground-based data. We present the distribution of galaxies with morphological type to H<23. We find relatively fewer irregular galaxies compared to an I-band sample from the Hubble Deep Field, which we attribute to their blue color, rather than to morphological K-corrections. We conclude that the irregulars are intrinsically faint blue galaxies at z<1.Comment: 13 pages, including 4 figures. Accepted for publication in ApJ Letter

A quest for frustration driven distortion in Y2Mo2O7

We investigated the nature of the freezing in the geometrically frustrated Heisenberg spin-glass Y2Mo2O7 by measuring the temperature dependence of the static internal magnetic field distribution above the spin-glass temperature, Tg, using the muSR technique. The evolution of the field distribution cannot be explained by changes in the spin susceptibility alone and suggests a lattice deformation. This possibility is addressed by numerical simulations of the Heisenberg Hamiltonian with magneto-elastic coupling at T>0.Comment: 5 pages 4 figures. Accepted for publication in PR

Multifractal analysis of perceptron learning with errors

Random input patterns induce a partition of the coupling space of a perceptron into cells labeled by their output sequences. Learning some data with a maximal error rate leads to clusters of neighboring cells. By analyzing the internal structure of these clusters with the formalism of multifractals, we can handle different storage and generalization tasks for lazy students and absent-minded teachers within one unified approach. The results also allow some conclusions on the spatial distribution of cells.Comment: 11 pages, RevTex, 3 eps figures, version to be published in Phys. Rev. E 01Jan9