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We show that a divergence-free measure on the plane is a continuous sum of unit tangent vector fields on rectifiable Jordan curves. This loop decomposition is more precise than the general decomposition in elementary solenoids given by S.K. Smirnov, when applied to the planar case. The proof involves extending the Fleming-Rishel formula to homogeneous BV functions (in any dimension), and establishing for such functions approximate continuity of measure theoretic connected components of suplevel sets as functions of the level. We apply these results to inverse potential problems whose source term is the divergence of some unknown (vector-valued) measure. A prototypical case is that of inverse magnetization problems when magnetizations are modeled by R3-valued Borel measures. We investigate methods for recovering a magnetization {\mu} by penalizing its measure theoretic total variation norm (TV). In particular, we prove that if a magnetization is supported in a plane, then TV-regularization schemes always have a unique minimizer, even in the presence of noise. It is further shown thatTV-norm minimization (among magnetizations generating the same field) uniquely recovers planar magnetizations in the following cases: when the magnetization is carried by a collection of sufficiently separated line segments and a set that is purely 1-unrectifiable, or when a superset of the support is tree-like. We note that such magnetizations can be recovered via TV-regularization schemes in the zero noise limit, by taking the regularization parameter to zero. This suggests definitions of sparsity in the present infinite dimensional context, that generate results akin to compressed sensing, Comment: 40 pages

We develop a framework for obtaining linear programming bounds for spherical codes whose inner products belong to a prescribed subinterval $[\ell,s]$ of $[-1,1)$. An intricate relationship between Levenshtein-type upper bounds on cardinality of codes with inner products in $[\ell,s]$ and lower bounds on the potential energy (for absolutely monotone interactions) for codes with inner products in $[\ell,1)$ (when the cardinality of the code is kept fixed) is revealed and explained. Thereby, we obtain a new extension of Levenshtein bounds for such codes. The universality of our bounds is exhibited by a unified derivation and their validity for a wide range of codes and potential functions., Comment: 18 pages, 1 figure

We study minimum energy problems relative to the $\alpha$-Riesz kernel $|x-y|^{\alpha-n}$, $\alpha\in(0,2]$, over signed Radon measures $\mu$ on $\mathbb R^n$, $n\geqslant3$, associated with a generalized condenser $(A_1,A_2)$, where $A_1$ is a relatively closed subset of a domain $D$ and $A_2=\mathbb R^n\setminus D$. We show that, though $A_2\cap\mathrm{Cl}_{\mathbb R^n}A_1$ may have nonzero capacity, this minimum energy problem is uniquely solvable (even in the presence of an external field) if we restrict ourselves to $\mu$ with $\mu^+\leqslant\xi$, where a constraint $\xi$ is properly chosen. We establish the sharpness of the sufficient conditions on the solvability thus obtained, provide descriptions of the weighted $\alpha$-Riesz potentials of the solutions, single out their characteristic properties, and analyze their supports. The approach developed is mainly based on the establishment of an intimate relationship between the constrained minimum $\alpha$-Riesz energy problem over signed measures associated with $(A_1,A_2)$ and the constrained minimum $\alpha$-Green energy problem over positive measures carried by $A_1$. The results are illustrated by examples., Comment: 30 pages

We study the constrained minimum energy problem with an external field relative to the $\alpha$-Riesz kernel $|x-y|^{\alpha-n}$ of order $\alpha\in(0,n)$ for a generalized condenser $\mathbf A=(A_i)_{i\in I}$ in $\mathbb R^n$, $n\geqslant 3$, whose oppositely charged plates intersect each other over a set of zero capacity. Conditions sufficient for the existence of minimizers are found, and their uniqueness and vague compactness are studied. Conditions obtained are shown to be sharp. We also analyze continuity of the minimizers in the vague and strong topologies when the condenser and the constraint both vary, describe the weighted equilibrium vector potentials, and single out their characteristic properties. Our arguments are based particularly on the simultaneous use of the vague topology and a suitable semimetric structure on a set of vector measures associated with $\mathbf A$, and the establishment of completeness theorems for proper semimetric spaces. The results remain valid for the logarithmic kernel on $\mathbb R^2$ and $\mathbf A$ with compact $A_i$, $i\in I$. The study is illustrated by several examples., Comment: 3 figures

For a compact $ d $-dimensional rectifiable subset of $ \mathbb{R}^{p} $ we study asymptotic properties as $ N\to\infty $ of $N$-point configurations minimizing the energy arising from a Riesz $ s $-potential $ 1/r^s $ and an external field in the hypersingular case $ s\geq d$. Formulas for the weak$ ^* $ limit of normalized counting measures of such optimal point sets and the first-order asymptotic values of minimal energy are obtained. As an application, we derive a method for generating configurations whose normalized counting measures converge to a given absolutely continuous measure supported on a rectifiable subset of $ \mathbb{R}^{p} $. Results on separation and covering properties of discrete minimizers are given. Our theorems are illustrated with several numerical examples., Comment: Corrected typos; added references

There are many ways to generate a set of nodes on the sphere for use in a variety of problems in numerical analysis. We present a survey of quickly generated point sets on $\mathbb{S}^2$, examine their equidistribution properties, separation, covering, and mesh ratio constants and present a new point set, equal area icosahedral points, with low mesh ratio. We analyze numerically the leading order asymptotics for the Riesz and logarithmic potential energy for these configurations with total points $N, 48 pages, 29 figures

Contains fulltext : 91833.pdf (Author’s version preprint ) (Open Access)

Minimum Riesz energy problems in the presence of an external field are analyzed for a condenser with touching plates. We obtain sufficient and/or necessary conditions for the solvability of these problems in both the unconstrained and the constrained settings, investigate the properties of minimizers, and prove their uniqueness. Furthermore, characterization theorems in terms of variational inequalities for the weighted potentials are established. The results obtained are illustrated by several examples., Comment: 32 pages, 1 figure

We derive and investigate lower bounds for the potential energy of finite spherical point sets (spherical codes). Our bounds are optimal in the following sense -- they cannot be improved by employing polynomials of the same or lower degrees in the Delsarte-Yudin method. However, improvements are sometimes possible and we provide a necessary and sufficient condition for the existence of such better bounds. All our bounds can be obtained in a unified manner that does not depend on the potential function, provided the potential is given by an absolutely monotone function of the inner product between pairs of points, and this is the reason for us to call them universal. We also establish a criterion for a given code of dimension $n$ and cardinality $N$ not to be LP-universally optimal, e.g. we show that two codes conjectured by Ballinger et al to be universally optimal are not LP-universally optimal., Comment: 25 pages, 3 figures, 3 tables

Let $A$ be a compact $d$-rectifiable set embedded in Euclidean space $\RR^p$, $d\le p$. For a given continuous distribution $\sigma(x)$ with respect to $d$-dimensional Hausdorff measure on $A$, our earlier results provided a method for generating $N$-point configurations on $A$ that have asymptotic distribution $\sigma (x)$ as $N\to \infty$; moreover such configurations are "quasi-uniform" in the sense that the ratio of the covering radius to the separation distance is bounded independent of $N$. The method is based upon minimizing the energy of $N$ particles constrained to $A$ interacting via a weighted power law potential $w(x,y)|x-y|^{-s}$, where $s>d$ is a fixed parameter and $w(x,y)=\left(\sigma(x)\sigma(y)\right)^{-({s}/{2d})}$. Here we show that one can generate points on $A$ with the above mentioned properties keeping in the energy sums only those pairs of points that are located at a distance of at most $r_N=C_N N^{-1/d}$ from each other, with $C_N$ being a positive sequence tending to infinity arbitrarily slowly. To do this we minimize the energy with respect to a varying truncated weight $v_N(x,y)=\Phi\(\left|x-y\right|/r_N\)w(x,y)$, where $\Phi:(0,\infty)\to [0,\infty)$ is a bounded function with $\Phi(t)=0$, $t\geq 1$, and $\lim_{t\to 0^+}\Phi(t)=1$. This reduces, under appropriate assumptions, the complexity of generating $N$ point `low energy' discretizations to order $N C_N^d$ computations.

For $N$-point best-packing configurations $\omega_N$ on a compact metric space $(A,\rho)$, we obtain estimates for the mesh-separation ratio $\gamma(\omega_N,A)$, which is the quotient of the covering radius of $\omega_N$ relative to $A$ and the minimum pairwise distance between points in $\omega_N$. For best-packing configurations $\omega_N$ that arise as limits of minimal Riesz $s$-energy configurations as $s\to \infty$, we prove that $\gamma(\omega_N,A)\le 1$ and this bound can be attained even for the sphere. In the particular case when N=5 on $S^2$ with $\rho$ the Euclidean metric, we prove our main result that among the infinitely many 5-point best-packing configurations there is a unique configuration, namely a square-base pyramid $\omega_5^*$, that is the limit (as $s\to \infty$) of 5-point $s$-energy minimizing configurations. Moreover, $\gamma(\omega_5^*,S^2)=1$.

We prove a conjecture of Ambrus, Ball and Erd\'{e}lyi that equally spaced points maximize the minimum of discrete potentials on the unit circle whenever the potential is of the form \sum_{k=1}^n f(d(z,z_k)), where $f:[0,\pi]\to [0,\infty]$ is non-increasing and strictly convex and $d(z,w)$ denotes the geodesic distance between $z$ and $w$ on the circle., Comment: 8 pages, 1 figure, to appear Discrete and Computational Geometry

While it is generally accepted that Type Ia supernovae are the result of the explosion of a carbon-oxygen White Dwarf accreting mass in a binary system, the details of their genesis still elude us, and the nature of the binary companion is uncertain. Kasen (2010) points out that the presence of a non-degenerate companion in the progenitor system could leave an observable trace: a flux excess in the early rise portion of the lightcurve caused by the ejecta impact with the companion itself. This excess would be observable only under favorable viewing angles, and its intensity depends on the nature of the companion. We searched for the signature of a non-degenerate companion in three years of Supernova Legacy Survey data by generating synthetic lightcurves accounting for the effects of shocking and comparing true and synthetic time series with Kolmogorov-Smirnov tests. Our most constraining result comes from noting that the shocking effect is more prominent in rest-frame B than V band: we rule out a contribution from white dwarf-red giant binary systems to Type Ia supernova explosions greater than 10% at 2 sigma, and than 20% at 3 sigma level., Comment: 14 pages, 15 figures, resubmitted to ApJ, figure 15 modified

(Abridged) Precision cosmology with Type Ia supernovae (SNe Ia) makes use of the fact that SN Ia luminosities depend on their light-curve shapes and colours. Using Supernova Legacy Survey (SNLS) and other data, we show that there is an additional dependence on the global characteristics of their host galaxies: events of the same light-curve shape and colour are, on average, 0.08mag (~4.0sigma) brighter in massive host galaxies (presumably metal-rich) and galaxies with low specific star-formation rates (sSFR). SNe Ia in galaxies with a low sSFR also have a smaller slope ("beta") between their luminosities and colours with ~2.7sigma significance, and a smaller scatter on SN Ia Hubble diagrams (at 95% confidence), though the significance of these effects is dependent on the reddest SNe. SN Ia colours are similar between low-mass and high-mass hosts, leading us to interpret their luminosity differences as an intrinsic property of the SNe and not of some external factor such as dust. If the host stellar mass is interpreted as a metallicity indicator, the luminosity trends are in qualitative agreement with theoretical predictions. We show that the average stellar mass, and therefore the average metallicity, of our SN Ia host galaxies decreases with redshift. The SN Ia luminosity differences consequently introduce a systematic error in cosmological analyses, comparable to the current statistical uncertainties on parameters such as w. We show that the use of two SN Ia absolute magnitudes, one for events in high-mass (metal-rich) galaxies, and one for events in low-mass (metal-poor) galaxies, adequately corrects for the differences. Cosmological fits incorporating these terms give a significant reduction in chi^2 (3.8-4.5sigma). We conclude that future SN Ia cosmological analyses should use a correction of this (or similar) form to control demographic shifts in the galaxy population., Comment: Accepted for publication in MNRAS.

We derive the complete asymptotic expansion in terms of powers of $N$ for the geodesic $f$-energy of $N$ equally spaced points on a rectifiable simple closed curve $\Gamma$ in ${\mathbb R}^p$, $p\geq2$, as $N \to \infty$. For $f$ decreasing and convex, such a point configuration minimizes the $f$-energy $\sum_{j\neq k}f(d(\mathbf{x}_j, \mathbf{x}_k))$, where $d$ is the geodesic distance (with respect to $\Gamma$) between points on $\Gamma$. Completely monotonic functions, analytic kernel functions, Laurent series, and weighted kernel functions $f$ are studied. % Of particular interest are the geodesic Riesz potential $1/d^s$ ($s \neq 0$) and the geodesic logarithmic potential $\log(1/d)$. By analytic continuation we deduce the expansion for all complex values of $s$.

We have combined the large SN Ia database of the Canada-France-Hawaii Telescope Supernova Legacy Survey and catalogs of galaxies with photometric redshifts, VLA 1.4 GHz radio sources, and Spitzer infrared sources. We present eight SNe Ia in early-type host galaxies which have counterparts in the radio and infrared source catalogs. We find the SN Ia rate in subsets of radio and infrared early-type galaxies is ~1-5 times the rate in all early-type galaxies, and that any enhancement is always, Comment: 14 pages, 6 figures, 7 tables, accepted for publication in AJ

Let $A$ be a compact set in ${\mathbb R}^p$ of Hausdorff dimension $d$. For $s\in(0,d)$, the Riesz $s$-equilibrium measure $\mu^s$ is the unique Borel probability measure with support in $A$ that minimizes $$ I_s(\mu):=\iint\frac{1}{|x-y|^s}d\mu(y)d\mu(x)$$ over all such probability measures. If $A$ is strongly $({\mathcal H}^d, d)$-rectifiable, then $\mu^s$ converges in the weak-star topology to normalized $d$-dimensional Hausdorff measure restricted to $A$ as $s$ approaches $d$ from below.

We present a measurement of the distant Type la supernova (SN Ia) rate derived from the first 2 yr of the Canada-France-Hawaii Telescope Supernova Legacy Survey. We observed four 1° × 1° fields with a typical temporal frequency of (Δt) ∼ 4 observer-frame days over time spans of 158-211 days per season for each field, with breaks during the full Moon. We used 8-10 m class telescopes for spectroscopic follow-up to confirm our candidates and determine their redshifts. Our starting sample consists of 73 spectroscopically verified SNe la in the redshift range 0.2 < z < 0.6. We derive a volumetric SN Ia rate of r V(〈z〉 = 0.47) = [0.42 0.090.13(syst.) ± 0.06(stat.)] × 10 -4 yr -1 Mpc 3, assuming h = 0.7, Ω m -0.3, and a flat cosmology. Using recently published galaxy luminosity functions derived in our redshift range, we derive a SN Ia rate per unit luminosity of r L(〈z〉 = 0.47) = 0.154 -0.0330.048(syst.) 0.0310.039(stat.) SN units. Using our rate alone, we place an upper limit on the component of SN la production that tracks the cosmic star formation history of 1 SN la per 10 3 M ⊙ of stars formed. Our rate and other rates from surveys using spectroscopic sample confirmation display only a modest evolution out to z = 0.55. © 2006. The American Astronomical Society. All rights reserved.

We present distance measurements to 71 high redshift type Ia supernovae discovered during the first year of the 5-year Supernova Legacy Survey (SNLS). These events were detected and their multi-color light-curves measured using the MegaPrime/MegaCam instrument at the Canada-France-Hawaii Telescope (CFHT), by repeatedly imaging four one-square degree fields in four bands. Follow-up spectroscopy was performed at the VLT, Gemini and Keck telescopes to confirm the nature of the supernovae and to measure their redshift. With this data set, we have built a Hubble diagram extending to z=1, with all distance measurements involving at least two bands. Systematic uncertainties are evaluated making use of the multi-band photometry obtained at CFHT. Cosmological fits to this first year SNLS Hubble diagram give the following results : Omega_M = 0.263 +/- 0.042(stat) +/- 0.032(sys) for a flat LambdaCDM model; and w = -1.023 +/- 0.090(stat) +/- 0.054(sys) for a flat cosmology with constant equation of state w when combined with the constraint from the recent Sloan Digital Sky Survey measurement of baryon acoustic oscillations., Comment: (The SNLS Collaboration) 24 pages, 13 figures, Accepted in A&A. Computer readable tables at http://snls.in2p3.fr/conf/papers/cosmo1/

We report measurements of $��_M$, $��_��$, and w from eleven supernovae at z=0.36-0.86 with high-quality lightcurves measured using WFPC-2 on the HST. This is an independent set of high-redshift supernovae that confirms previous supernova evidence for an accelerating Universe. Combined with earlier Supernova Cosmology Project data, the new supernovae yield a flat-universe measurement of the mass density $��_M=0.25^{+0.07}_{-0.06}$ (statistical) $\pm0.04$ (identified systematics), or equivalently, a cosmological constant of $��_��=0.75^{+0.06}_{-0.07}$ (statistical) $\pm0.04$ (identified systematics). When the supernova results are combined with independent flat-universe measurements of $��_M$ from CMB and galaxy redshift distortion data, they provide a measurement of $w=-1.05^{+0.15}_{-0.20}$ (statistical) $\pm0.09$ (identified systematic), if w is assumed to be constant in time. The new data offer greatly improved color measurements of the high-redshift supernovae, and hence improved host-galaxy extinction estimates. These extinction measurements show no anomalous negative E(B-V) at high redshift. The precision of the measurements is such that it is possible to perform a host-galaxy extinction correction directly for individual supernovae without any assumptions or priors on the parent E(B-V) distribution. Our cosmological fits using full extinction corrections confirm that dark energy is required with $P(��_��>0)>0.99$, a result consistent with previous and current supernova analyses which rely upon the identification of a low-extinction subset or prior assumptions concerning the intrinsic extinction distribution., Accepted for publication in Astrophysical Journal; For data tables and full-resolution figures, see http://supernova.lbl.gov

We present an analysis of the light curves of 9.1 million stars observed during three seasons by EROS (Experience de Recherche d'Objets Sombres), in the Galactic plane away from the bulge. Seven stars exhibit luminosity variations compatible with gravitational microlensing effects due to unseen objects. The corresponding optical depth, averaged over four directions, is tau = 0.45 +0.24 -0.11 x 10^-6. While this value is compatible with expectations from simple galactic models under reasonable assumptions on the target distances, we find an excess of events with short timescale towards the direction closest to the Galactic Centre., Comment: 4 pages, 3 figures, submitted for publication in A&A Letters

We report on intensive photometric monitoring on 18 June 1998 of MACHO SMC-98-1, a binary-lens microlensing event seen toward the Small Magellanic Cloud (SMC). The observations cover 5.3 hours (UT 5:17 -- 10:37), and show a sharp drop of 1.8 mag during the first 1.8 hours, followed by an abrupt flattening at UT 7:08 +- 0:02. We interpret the kink at 7:08 as the end of the second caustic crossing (when the source first moved completely outside the caustic). These results indicate that mu sin(phi), Comment: 4 pages, 4 ps figures. Submitted to Astronomy & Astrophysics. Halo interpretation excluded at the 93% CL (instead of 89%)

Recent surveys monitoring millions of light curves of resolved stars in the LMC have discovered several microlensing events. Unresolved stars could however significantly contribute to the microlensing rate towards the LMC. Monitoring pixels, as opposed to individual stars, should be able to detect stellar variability as a variation of the pixel flux. We present a first application of this new type of analysis (Pixel Method) to the LMC Bar. We describe the complete procedure applied to the EROS 91-92 data (one tenth of the existing CCD data set) in order to monitor pixel fluxes. First, geometric and photometric alignments are applied to each images. Averaging the images of each night reduces significantly the noise level. Second, one light curve for each of the 2.1 10^6 pixels is built and pixels are lumped into 3.6"x3.6" super-pixels, one for each elementary pixel. An empirical correction is then applied to account for seeing variations. We find that the final super-pixel light curves fluctuate at a level of 1.8% of the flux in blue and 1.3% in red. We show that this noise level corresponds to about twice the expected photon noise and confirms previous assumptions used for the estimation of the contribution of unresolved stars. We also demonstrate our ability to correct very efficiently for seeing variations affecting each pixel flux. The technical results emphasised here show the efficacy of the Pixel Method and allow us to study luminosity variations due to possible microlensing events and variable stars in two companion papers., Comment: 15 pages, 26 figures -- accepted for publication in Astronomy and Astrophysics Supplement Series -- Changed contents + Corrected typos

We apply the pixel method of analysis (sometimes called ``pixel lensing'') to a small subset of the EROS-1 microlensing observations of the bar of the Large Magellanic Cloud (LMC). The pixel method is designed to find microlensing events of unresolved source stars and had heretofore been applied only to M31 where essentially all sources are unresolved. With our analysis optimised for the detection of long-duration microlensing events due to 0.01-1 Mo Machos, we detect no microlensing events and compute the corresponding detection efficiencies. We show that the pixel method should detect 10 to 20 times more microlensing events for M>0.05 Mo Machos compared to a classical analysis of the same data which latter monitors only resolved stars. In particular, we show that for a full halo of Machos in the mass range 0.1 -- 0.5 Mo, a pixel analysis of the three-year EROS-1 data set covering 0.39 deg^2 would yield 4 events. We apply the pixel method of analysis (sometimes called ''pixel lensing'') to a small subset of the EROS-1 microlensing observations of the bar of the Large Magellanic Cloud (LMC). The pixel method is designed to find microlensing events of unresolved source stars and had heretofore been applied only to M31 where essentially all sources are unresolved. With our analysis optimised for the detection of long-duration microlensing events due to 0.01-1 Mo Machos, we detect no microlensing events and compute the corresponding detection efficiencies. We show that the pixel method, applied to crowded fields, should detect 10 to 20 times more microlensing events for M>0.05 Mo Machos compared to a classical analysis of the same data which latter monitors only resolved stars. In particular, we show that for a full halo of Machos in the mass range 0.1-0.5 M $\bigodot$, a pixel analysis of the three-year EROS-1 data set covering $0.39deg^{2}$ would yield $\simeq 4$ events.

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We present the Period-Luminosity relations from 290 Cepheids towards the LMC and 590 Cepheids towards the SMC. The two data sets were obtained using the two wide field CCD cameras of the EROS 2 microlensing survey. We observe a significant slope change of the period-luminosity relation for the SMC fundamental mode Cepheids with periods shorter than 2 days. Many possible experimental biases have been investigated, but none can account for this effect. We also observe different spatial distributions for SMC Cepheids with different ages i.e. periods. Different possible explanations of the slope change are discussed.

We present the analysis of the light curves of 8.5 million stars observed during two seasons by EROS (Experience de Recherche d'Objets Sombres), in the galactic plane away from the bulge. Three stars have been found that exhibit luminosity variations compatible with gravitational microlensing effects due to unseen objects. The corresponding optical depth, averaged over four directions, is 0.38 (+0.53, -0.15) 10^{-6}. All three candidates have long Einstein radius crossing times ($\sim$ 70 to 100 days). For one of them, the lack of evidence for a parallax or a source size effect enabled us to constrain the lens-source % geometric configuration. Another candidate displays a modulation of the magnification, which is compatible with the lensing of a binary source. The interpretation of the optical depths inferred from these observations is hindered by the imperfect knowledge of the distance to the target stars. Our measurements are compatible with expectations from simple galactic models under reasonable assumptions on the target distances., Comment: 11 pages, 13 figures, accepted by A&A in Aug 99

We report the discovery of two L dwarfs (the new spectral class defined for dwarfs cooler than the M type) in a two-epoch CCD proper motion survey of 413 square degrees, complemented by infrared photometry from DENIS. One of them has a strong lithium line and is therefore a brown dwarf. The other is a common proper motion companion to the mid-M dwarf LHS 102 (GJ 1001), which has a well determined trigonometric parallax. LHS 102B is thus the coolest L dwarf of known distance and luminosity. Its infrared absolute photometry are very well reproduced by the Allard et al DUSTY models.