Single Machine Scheduling with Release Dates

We consider the scheduling problem of minimizing the average weighted completion time of n jobs with release dates on a single machine. We first study two linear programming relaxations of the problem, one based on a time-indexed formulation, the other on a completiontime formulation. We show their equivalence by proving that a O(n log n) greedy algorithm leads to optimal solutions to both relaxations. The proof relies on the notion of mean busy times of jobs, a concept which enhances our understanding of these LP relaxations. Based on the greedy solution, we describe two simple randomized approximation algorithms, which are guaranteed to deliver feasible schedules with expected objective value within factors of 1.7451 and 1.6853, respectively, of the optimum. They are based on the concept of common and independent a-points, respectively. The analysis implies in particular that the worst-case relative error of the LP relaxations is at most 1.6853, and we provide instances showing that it is at least e/(e - 1) 1.5819. Both algorithms may be derandomized, their deterministic versions running in O(n2 ) time. The randomized algorithms also apply to the on-line setting, in which jobs arrive dynamically over time and one must decide which job to process without knowledge of jobs that will be released afterwards

Temperature effects on dislocation core energies in silicon and germanium

Temperature effects on the energetics of the 90-degree partial dislocation in silicon and germanium are investigated, using non-equilibrium methods to estimate free energies, coupled with Monte Carlo simulations. Atomic interactions are described by Tersoff and EDIP interatomic potentials. Our results indicate that the vibrational entropy has the effect of increasing the difference in free energy between the two possible reconstructions of the 90-degree partial, namely, the single-period and the double-period geometries. This effect further increases the energetic stability of the double-period reconstruction at high temperatures. The results also indicate that anharmonic effects may play an important role in determining the structural properties of these defects in the high-temperature regime.Comment: 8 pages in two-column physical-review format with six figure

From the Mendeleev periodic table to particle physics and back to the periodic table

We briefly describe in this paper the passage from Mendeleev's chemistry (1869) to atomic physics (in the 1900's), nuclear physics (in the 1932's) and particle physics (from 1953 to 2006). We show how the consideration of symmetries, largely used in physics since the end of the 1920's, gave rise to a new format of the periodic table in the 1970's. More specifically, this paper is concerned with the application of the group SO(4,2)xSU(2) to the periodic table of chemical elements. It is shown how the Madelung rule of the atomic shell model can be used for setting up a periodic table that can be further rationalized via the group SO(4,2)xSU(2) and some of its subgroups. Qualitative results are obtained from this nonstandard table.Comment: 15 pages; accepted for publication in Foundations of Chemistry (special issue to commemorate the one hundredth anniversary of the death of Mendeleev who died in 1907); version 2: 16 pages; some sentences added; acknowledgment and references added; misprints correcte

Extended Weyl Calculus and Application to the Phase-Space Schr\"{o}dinger Equation

We show that the Schr\"{o}dinger equation in phase space proposed by Torres-Vega and Frederick is canonical in the sense that it is a natural consequence of the extended Weyl calculus obtained by letting the Heisenberg group act on functions (or half-densities) defined on phase space. This allows us, in passing, to solve rigorously the TF equation for all quadratic Hamiltonians.Comment: To appear in J. Phys. A: Math. and genera

Neutral material around the B[e] supergiant star LHA 115-S 65: An outflowing disk or a detached Keplerian rotating disk?

B[e] supergiants are surrounded by large amounts of hydrogen neutral material, traced by the emission in the optical [OI] lines. This neutral material is most plausibly located within their dense, cool circumstellar disks, which are formed from the (probably non-spherically symmetric) wind material released by the star. Neither the formation mechanism nor the resulting structure and internal kinematics of these disks (or disk-like outflows) are well known. However, rapid rotation, lifting the material from the equatorial surface region, seems to play a fundamental role. The B[e] supergiant LHA 115-S 65 (S65) in the SMC is one of the two most rapidly rotating B[e] stars known. Its almost edge-on orientation allows a detailed kinematical study of its optically thin forbidden emission lines. With a focus on the [OI] lines, we test the two plausible disk scenarios: the outflowing and the Keplerian rotating disk. Based on high- and low-resolution optical spectra, we investigate the density and temperature structure in those disk regions that are traced by the [OI] emission to constrain the disk sizes and mass fluxes needed to explain the observed [OI] line luminosities. In addition, we compute the emerging line profiles expected for either an outflowing disk or a Keplerian rotating disk, which can directly be compared to the observed profiles. Both disk scenarios deliver reasonably good fits to the line luminosities and profiles of the [OI] lines. Nevertheless, the Keplerian disk model seems to be the more realistic one, because it also agrees with the kinematics derived from the large number of additional lines in the spectrum. As additional support for the presence of a high-density, gaseous disk, the spectrum shows two very intense and clearly double-peaked [CaII] lines. We discuss a possible disk-formation mechanism, and similarities between S65 and the group of LBVs.Comment: 13 pages, 12 figures, accepted for publication in A&

Galactic Bulge Microlensing Optical Depth from EROS-2

We present a new EROS-2 measurement of the microlensing optical depth toward the Galactic Bulge. Light curves of $5.6\times 10^{6}$ clump-giant stars distributed over $66 \deg^2$ of the Bulge were monitored during seven Bulge seasons. 120 events were found with apparent amplifications greater than 1.6 and Einstein radius crossing times in the range 5 {\rm d}. This is the largest existing sample of clump-giant events and the first to include northern Galactic fields. In the Galactic latitude range 1.4\degr<|b|<7.0\degr, we find $\tau/10^{-6}=(1.62 \pm 0.23)\exp[-a(|b|-3 {\rm deg})]$ with $a=(0.43 \pm0.16)\deg^{-1}$. These results are in good agreement with our previous measurement, with recent measurements of the MACHO and OGLE-II groups, and with predictions of Bulge models.Comment: accepted A&A, minor revision

Observation of periodic variable stars towards the galactic spiral arms by EROS II

We present the results of a massive variability search based on a photometric survey of a six square degree region along the Galactic plane at ($l = 305^\circ$, $b = -0.8^\circ$) and ($l = 330^\circ$, $b = -2.5^\circ$). This survey was performed in the framework of the EROS II (Exp\'erience de Recherche d'Objets Sombres) microlensing program. The variable stars were found among 1,913,576 stars that were monitored between April and June 1998 in two passbands, with an average of 60 measurements. A new period-search technique is proposed which makes use of a statistical variable that characterizes the overall regularity of the flux versus phase diagram. This method is well suited when the photometric data are unevenly distributed in time, as is our case. 1,362 objects whose luminosity varies were selected. Among them we identified 9 Cepheids, 19 RR Lyrae, 34 Miras, 176 eclipsing binaries and 266 Semi-Regular stars. Most of them are newly identified objects. The cross-identification with known catalogues has been performed. The mean distance of the RR Lyrae is estimated to be $\sim 4.9 \pm 0.3$ kpc undergoing an average absorption of $\sim 3.4 \pm 0.2$ magnitudes. This distance is in good agreement with the one of disc stars which contribute to the microlensing source star population.Our catalogue and light curves are available electronically from the CDS, Strasbourg and from our Web site http://eros.in2p3.fr.Comment: 15 pages, 11 figures, accepted in A&A (april 2002

The EROS2 search for microlensing events towards the spiral arms: the complete seven season results

The EROS-2 project has been designed to search for microlensing events towards any dense stellar field. The densest parts of the Galactic spiral arms have been monitored to maximize the microlensing signal expected from the stars of the Galactic disk and bulge. 12.9 million stars have been monitored during 7 seasons towards 4 directions in the Galactic plane, away from the Galactic center. A total of 27 microlensing event candidates have been found. Estimates of the optical depths from the 22 best events are provided. A first order interpretation shows that simple Galactic models with a standard disk and an elongated bulge are in agreement with our observations. We find that the average microlensing optical depth towards the complete EROS-cataloged stars of the spiral arms is $\bar{\tau} =0.51\pm .13\times 10^{-6}$, a number that is stable when the selection criteria are moderately varied. As the EROS catalog is almost complete up to $I_C=18.5$, the optical depth estimated for the sub-sample of bright target stars with $I_C<18.5$ ($\bar{\tau}=0.39\pm >.11\times 10^{-6}$) is easier to interpret. The set of microlensing events that we have observed is consistent with a simple Galactic model. A more precise interpretation would require either a better knowledge of the distance distribution of the target stars, or a simulation based on a Galactic model. For this purpose, we define and discuss the concept of optical depth for a given catalog or for a limiting magnitude.Comment: 22 pages submitted to Astronomy & Astrophysic

New Magellanic Cloud R Coronae Borealis and DY Per type stars from the EROS-2 database: the connection between RCBs, DYPers and ordinary carbon stars

R Coronae Borealis stars (RCB) are a rare type of evolved carbon-rich supergiant stars that are increasingly thought to result from the merger of two white dwarfs, called the Double degenerate scenario. This scenario is also studied as a source, at higher mass, of type Ia Supernovae (SnIa) explosions. Therefore a better understanding of RCBs composition would help to constrain simulations of such events. We searched for and studied RCB stars in the EROS Magellanic Clouds database. We also extended our research to DY Per type stars (DYPers) that are expected to be cooler RCBs (T~3500 K) and much more numerous than their hotter counterparts. The light curves of ~70 millions stars have been analysed to search for the main signature of RCBs and DYPers: a large drop in luminosity. Follow-up optical spectroscopy was used to confirm each photometric candidate found. We have discovered and confirmed 6 new Magellanic Cloud RCB stars and 7 new DYPers, but also listed new candidates: 3 RCBs and 14 DYPers. We estimated a range of Magellanic RCB shell temperatures between 360 and 600 K. We confirm the wide range of absolute luminosity known for RCB stars, M_V~-5.2 to -2.6. Our study further shows that mid-infrared surveys are ideal to search for RCB stars, since they have thinner and cooler circumstellar shells than classical post-AGB stars. In addition, by increasing the number of known DYPers by ~400%, we have been able to shed light on the similarities in the spectral energy distribution between DYPers and ordinary carbon stars. We also observed that DYPer circumstellar shells are fainter and hotter than those of RCBs. This suggests that DYPers may simply be ordinary carbon stars with ejection events, but more abundance analysis is necessary to give a status on a possible evolutionnary connexion between RCBs and DYPers.Comment: 22 pages, 38 figures, Accepted for publication in A&