Cubic Curves, Finite Geometry and Cryptography

Some geometry on non-singular cubic curves, mainly over finite fields, is surveyed. Such a curve has 9,3,1 or 0 points of inflexion, and cubic curves are classified accordingly. The group structure and the possible numbers of rational points are also surveyed. A possible strengthening of the security of elliptic curve cryptography is proposed using a `shared secret' related to the group law. Cubic curves are also used in a new way to construct sets of points having various combinatorial and geometric properties that are of particular interest in finite Desarguesian planes.Comment: This is a version of our article to appear in Acta Applicandae Mathematicae. In this version, we have corrected a sentence in the third paragraph. The final publication is available at springerlink.com at http://www.springerlink.com/content/xh85647871215644

HST Snaphot Study of Variable Stars in Globular Clusters: Inner Region of NGC 6441

[Abridged] We present the results of a Hubble Space Telescope snapshot program to survey the inner region of the globular cluster NGC 6441 for its variable stars. A total of 57 variable stars was found including 38 RR Lyrae stars, 6 Population II Cepheids, and 12 long period variables. Of the RR Lyrae stars observed in this survey, 26 are pulsating in the fundamental mode with a mean period of 0.753d and 12 are first-overtone mode pulsators with a mean period of 0.365d. These values match up very well with those found in ground-based surveys. Combining all the available data for NGC 6441, we find mean periods of 0.759d and 0.375d for the RRab and RRc stars, respectively. We also find that the RR Lyrae in this survey are located in the same regions of a period-amplitude diagram as those found in ground-based surveys. Although NGC 6441 is a metal-rich globular cluster, its RR Lyrae more closely resemble those in Oosterhoff type II globular clusters. However, even compared to typical Oosterhoff type II systems, the mean period of its RRab stars is unusually long. We also derived I-band period-luminosity relations for the RR Lyrae stars. Of the six Population II Cepheids, five are of W Virginis type and one is a BL Herculis variable stars. This makes NGC 6441, along with NGC 6388, the most metal-rich globular cluster known to contain these types of variable stars. Another variable, V118, may also be a Population II Cepheid given its long period and its separation in magnitude from the RR Lyrae stars. We argue that there does not appear to be a change in the period-luminosity relation slope between the BL Herculis and W Virginis stars, but that a change of slope does occur when the RV Tauri stars are added to the period-luminosity relation.Comment: 28 pages, including 9 figures and 8 tables, emulateapj5/apjfonts style. Accepted by the Astronomical Journal. Approximate publication date September 2003. We recommend the interested reader to download the preprint with full-resolution figures, which can be found at http://www.astro.puc.cl/~mcatelan/Pritzl.zi

The Complete Solution of 2D Superfield Supergravity from graded Poisson-Sigma Models and the Super Pointparticle

Recently an alternative description of 2d supergravities in terms of graded Poisson-Sigma models (gPSM) has been given. As pointed out previously by the present authors a certain subset of gPSMs can be interpreted as "genuine" supergravity, fulfilling the well-known limits of supergravity, albeit deformed by the dilaton field. In our present paper we show that precisely that class of gPSMs corresponds one-to-one to the known dilaton supergravity superfield theories presented a long time ago by Park and Strominger. Therefore, the unique advantages of the gPSM approach can be exploited for the latter: We are able to provide the first complete classical solution for any such theory. On the other hand, the straightforward superfield formulation of the point particle in a supergravity background can be translated back into the gPSM frame, where "supergeodesics" can be discussed in terms of a minimal set of supergravity field degrees of freedom. Further possible applications like the (almost) trivial quantization are mentioned.Comment: 48 pages, 1 figure. v3: after final version, typos correcte

OPE's and the Dilaton Beta-Function for the 2-D N=1 Supersymmetric Non-Linear σ\sigma-Model

Using the superspace formalism, we compute for the two-dimensional N=1 supersymmetric non-linear $\sigma$-model, the order $(\alpha^{\prime})^{2}$ $(R_{mnpq})^2$ (three-loop) correction to the central charge via the operator product expansion of the supercurrent with itself. The contribution vanishes, in agreement with previous results obtained from the usual $\sigma$-model $\beta$-function approach.Comment: 10 pp. + 1 uuencoded figur

Horizontal Branch Stars: The Interplay between Observations and Theory, and Insights into the Formation of the Galaxy

We review HB stars in a broad astrophysical context, including both variable and non-variable stars. A reassessment of the Oosterhoff dichotomy is presented, which provides unprecedented detail regarding its origin and systematics. We show that the Oosterhoff dichotomy and the distribution of globular clusters (GCs) in the HB morphology-metallicity plane both exclude, with high statistical significance, the possibility that the Galactic halo may have formed from the accretion of dwarf galaxies resembling present-day Milky Way satellites such as Fornax, Sagittarius, and the LMC. A rediscussion of the second-parameter problem is presented. A technique is proposed to estimate the HB types of extragalactic GCs on the basis of integrated far-UV photometry. The relationship between the absolute V magnitude of the HB at the RR Lyrae level and metallicity, as obtained on the basis of trigonometric parallax measurements for the star RR Lyrae, is also revisited, giving a distance modulus to the LMC of (m-M)_0 = 18.44+/-0.11. RR Lyrae period change rates are studied. Finally, the conductive opacities used in evolutionary calculations of low-mass stars are investigated. [ABRIDGED]Comment: 56 pages, 22 figures. Invited review, to appear in Astrophysics and Space Scienc

Variable Stars in Galactic Globular Clusters

Based on a search of the literature up to May 2001, the number of known variable stars in Galactic globular clusters is approximately 3000. Of these, more than 2200 have known periods and the majority (approximately 1800) are of the RR Lyrae type. In addition to the RR Lyrae population, there are approximately 100 eclipsing binaries, 120 SX Phe variables, 60 Cepheids (including population II Cepheids, anomalous Cepheids and RV Tauri) and 120 SR/red variables. The mean period of the fundamental mode RR Lyrae variables is 0.585, for the overtone variables it is 0.342 (0.349 for the first-overtone pulsators and 0.296 for the second-overtone pulsators) and approximately 30% are overtone pulsators. These numbers indicate that about 65% of RR Lyrae variables in Galactic globular clusters belong to Oosterhoff type I systems. The mean period of the RR Lyrae variables in the Oosterhoff type I clusters seems to be correlated with metal abundance in the sense that the periods are longer in the more metal poor clusters. Such a correlation does not exist for the Oosterhoff type II clusters. Most of the Cepheids are in clusters with blue horizontal branches.Comment: 45 pages, 10 figures, to be published in AJ November 200