The lambda-dimension of commutative arithmetic rings

It is shown that every commutative arithmetic ring $R$ has $lambda$-dimension $leq 3$. An example of a commutative Kaplansky ring with $lambda$-dimension 3 is given. If $R$ satisfies an additional condition then $lambda$-dim($R$

Indecomposable modules and Gelfand rings

It is proved that a commutative ring is clean if and only if it is Gelfand with a totally disconnected maximal spectrum. Commutative rings for which each indecomposable module has a local endomorphism ring are studied. These rings are clean and elementary divisor rings

On the Link Between Strongly Connected Iteration Graphs and Chaotic Boolean Discrete-Time Dynamical Systems

Chaotic functions are characterized by sensitivity to initial conditions, transitivity, and regularity. Providing new functions with such properties is a real challenge. This work shows that one can associate with any Boolean network a continuous function, whose discrete-time iterations are chaotic if and only if the iteration graph of the Boolean network is strongly connected. Then, sufficient conditions for this strong connectivity are expressed on the interaction graph of this network, leading to a constructive method of chaotic function computation. The whole approach is evaluated in the chaos-based pseudo-random number generation context

Destriping of Polarized Data in a CMB Mission with a Circular Scanning Strategy

A major problem in Cosmic Microwave Background (CMB) anisotropy mapping, especially in a total-power mode, is the presence of low-frequency noise in the data streams. If unproperly processed, such low-frequency noise leads to striping in the maps. To deal with this problem, solutions have already been found for mapping the CMB temperature fluctuations but no solution has yet been proposed for the measurement of CMB polarization. Complications arise due to the scan-dependent orientation of the measured polarization. In this paper, we investigate a method for building temperature and polarization maps free of striping effects in the case of a circular scanning strategy mission such as that of the Planck satellite

About the connection between the CℓC_{\ell} power spectrum of the Cosmic Microwave Background and the Γm\Gamma_{m} Fourier spectrum of rings on the sky

In this article we present and study a scaling law of the $m\Gamma_m$ CMB Fourier spectrum on rings which allows us (i) to combine spectra corresponding to different colatitude angles (e.g. several detectors at the focal plane of a telescope), and (ii) to recover the $C_l$ power spectrum once the $\Gamma_m$ coefficients have been measured. This recovery is performed numerically below the 1% level for colatitudes $\Theta> 80^\circ$ degrees. In addition, taking advantage of the smoothness of the $C_l$ and of the $\Gamma_m$, we provide analytical expressions which allow to recover one of the spectrum at the 1% level, the other one being known.Comment: 8 pages, 8 figure

Analysis of CMB polarization on an incomplete sky

The full sky cosmic microwave background polarization field can be decomposed into 'electric' and 'magnetic' components. Working in harmonic space we construct window functions that allow clean separation of the electric and magnetic modes from observations over only a portion of the sky. Our construction is exact for azimuthally symmetric patches, but should continue to perform well for arbitrary patches. From the window functions we obtain variables that allow for robust estimation of the magnetic component without risk of contamination from the probably much larger electric signal. For isotropic, uncorrelated noise the variables have a very simple diagonal noise correlation, and further analysis using them should be no harder than analysing the temperature field. For an azimuthally-symmetric patch, such as that obtained from survey missions when the galactic region is removed, the exactly-separated variables are fast to compute allowing us to estimate the magnetic signal that could be detected by the Planck satellite in the absence of non-galactic foregrounds. We also discuss the sensitivity of future experiments to tensor modes in the presence of a magnetic signal generated by weak lensing, and give lossless methods for analysing the electric polarization field in the case that the magnetic component is negligible.Comment: 27 pages, 8 figures. New appendix on weak signal detection and revised plots using a better statistic. Other changes to match version accepted by PRD. Sample source code now available at http://cosmologist.info/pola

Scanning strategy for mapping the Cosmic Microwave Background anisotropies with Planck

We present simulations of different scanning strategies for the Planck satellite. We review the properties of slow- and fast-precession strategies in terms of uniformity of the integration time on the sky, the presence of low-redundancy areas, the presence of deep fields, the presence of sharp gradients in the integration time, and the redundancy of the scanning directions. We also compare the results obtained when co-adding all detectors of a given frequency channel. The slow-precession strategies allow a good uniformity of the coverage, while providing two deep fields. On the other hand, they do not allow a wide spread of the scan-crossing directions, which is a feature of the fast-precession strategies. However, the latter suffer from many sharp gradients and low-coverage areas on the sky. On the basis of these results, the strategy for Planck can be selected to be a slow (e.g. 4 month-period) sinusoidal or cycloidal scanning.Comment: 10 pages, 12 figures, accepted in A&A. Degraded JPEG figure

Observational Limits on Machos in the Galactic Halo

We present final results from the first phase of the EROS search for gravitational microlensing of stars in the Magellanic Clouds by unseen deflectors (machos: MAssive Compact Halo Objects). The search is sensitive to events with time scales between 15 minutes and 200 days corresponding to deflector masses in the range 1.e-7 to a few solar masses. Two events were observed that are compatible with microlensing by objects of mass of about 0.1 Mo. By comparing the results with the expected number of events for various models of the Galaxy, we conclude that machos in the mass range [1.e-7, 0.02] Mo make up less than 20% (95% C.L.) of the Halo dark matter.Comment: 4 pages, 3 Postscript figures, to be published in Astronomy & Astrophysic

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