A survey on signature-based Gr\"obner basis computations

This paper is a survey on the area of signature-based Gr\"obner basis algorithms that was initiated by Faug\`ere's F5 algorithm in 2002. We explain the general ideas behind the usage of signatures. We show how to classify the various known variants by 3 different orderings. For this we give translations between different notations and show that besides notations many approaches are just the same. Moreover, we give a general description of how the idea of signatures is quite natural when performing the reduction process using linear algebra. This survey shall help to outline this field of active research.Comment: 53 pages, 8 figures, 11 table

Predicting zero reductions in Gr\"obner basis computations

Since Buchberger's initial algorithm for computing Gr\"obner bases in 1965 many attempts have been taken to detect zero reductions in advance. Buchberger's Product and Chain criteria may be known the most, especially in the installaton of Gebauer and M\"oller. A relatively new approach are signature-based criteria which were first used in Faug\`ere's F5 algorithm in 2002. For regular input sequences these criteria are known to compute no zero reduction at all. In this paper we give a detailed discussion on zero reductions and the corresponding syzygies. We explain how the different methods to predict them compare to each other and show advantages and drawbacks in theory and practice. With this a new insight into algebraic structures underlying Gr\"obner bases and their computations might be achieved.Comment: 25 pages, 3 figure

Termination of Original F5

The original F5 algorithm introduced by Faug\`ere is formulated for any homogeneous polynomial set input. The correctness of output is shown for any input that terminates the algorithm, but the termination itself is proved only for the case of input being regular polynomial sequence. This article shows that algorithm correctly terminates for any homogeneous input without any reference to regularity. The scheme contains two steps: first it is shown that if the algorithm does not terminate it eventually generates two polynomials where first is a reductor for the second. But first step does not show that this reduction is permitted by criteria introduced in F5. The second step shows that if such pair exists then there exists another pair for which the reduction is permitted by all criteria. Existence of such pair leads to contradiction. Version v3 fixes the bibliography

Astrophysical Tests of Modified Gravity: A Screening Map of the Nearby Universe

Astrophysical tests of modified modified gravity theories in the nearby universe have been emphasized recently by Hui, Nicolis and Stubbs (2009) and Jain and VanderPlas (2011). A key element of such tests is the screening mechanism whereby general relativity is restored in massive halos or high density environments like the Milky Way. In chameleon theories of gravity, including all f(R) models, field dwarf galaxies may be unscreened and therefore feel an extra force, as opposed to screened galaxies. The first step to study differences between screened and unscreened galaxies is to create a 3D screening map. We use N-body simulations to test and calibrate simple approximations to determine the level of screening in galaxy catalogs. Sources of systematic errors in the screening map due to observational inaccuracies are modeled and their contamination is estimated. We then apply our methods to create a map out to 200 Mpc in the Sloan Digital Sky Survey footprint using data from the Sloan survey and other sources. In two companion papers this map will be used to carry out new tests of gravity using distance indicators and the disks of dwarf galaxies. We also make our screening map publicly available.Comment: 21 pages, 10 figure

Pulsating stars in NGC 6231 Frequency analysis and photometric mode identification near the main sequence

We used Johnson UBV photometric CCD observations to identify pulsating and other variable stars in the young open cluster NGC 6231. The multi-color information was used to classify pulsating variables, perform frequency analysis, and - where possible - to compare observed to theoretical amplitude ratios for mode identification. The data reduction was performed with standard IRAF tools. Differential light curves have been obtained by identifying a set of suitable comparison stars and the frequency analysis was then conducted on the basis of Fourier methods. Our classification of pulsating stars was based on the time scales and amplitudes of the variability with respect to the different filters and stellar parameters as calculated from published Str\"omgren and Geneva photometry. We identified 32 variable stars in the field of the cluster out of which 21 are confirmed members and twelve are newly detected variable stars. Ten stars were classified as Slowly Pulsating B (SPB) stars in NGC 6231 out of which seven are new discoveries. We also analyzed six previously reported {\beta} Cephei variables in more detail. One of them may be a hybrid {\beta} Cephei/SPB pulsator. In addition, we investigated five more previously suspected pulsators of this group which we cannot convincingly confirm. The remaining eleven variable stars are either not members of NGC 6231 or the membership status is questionable. Among them are three previously known {\delta} Scuti stars, two newly detected pulsators of this class, one new and two already known eclipsing binaries, one new SPB variable, one possible Pre-Main-Sequence (PMS) pulsator and another new variable star for which we cannot present a classification. With more than 20 main sequence pulsators of spectral type B, NGC 6231 becomes the open cluster with the largest population of such pulsating stars known.Comment: 27 pages, 35 figures, 3 Tables, accepted by A&A, abstract excessively shorted due to character limit

Quantum versus classical domains for teleportation with continuous variables

By considering the utilization of a classical channel without quantum entanglement, fidelity Fclassical=1/2 has been established as setting the boundary between classical and quantum domains in the teleportation of coherent states of the electromagnetic field [S. L. Braunstein, C. A. Fuchs, and H. J. Kimble, J. Mod. Opt. 47, 267 (2000)]. We further examine the quantum-classical boundary by investigating questions of entanglement and Bell-inequality violations for the Einstein-Podolsky-Rosen states relevant to continuous variable teleportation. The threshold fidelity for employing entanglement as a quantum resource in teleportation of coherent states is again found to be Fclassical=1/2. Likewise, violations of local realism onset at this same threshold, with the added requirement of overall efficiency η>2/3 in the unconditional case. By contrast, recently proposed criteria adapted from the literature on quantum-nondemolition detection are shown to be largely unrelated to the questions of entanglement and Bell-inequality violations