Splitting fields and general differential Galois theory

An algebraic technique is presented that does not use results of model theory and makes it possible to construct a general Galois theory of arbitrary nonlinear systems of partial differential equations. The algebraic technique is based on the search for prime differential ideals of special form in tensor products of differential rings. The main results demonstrating the work of the technique obtained are the theorem on the constructedness of the differential closure and the general theorem on the Galois correspondence for normal extensions..Comment: 33 pages, this version coincides with the published on

Random Networks Tossing Biased Coins

In statistical mechanical investigations on complex networks, it is useful to employ random graphs ensembles as null models, to compare with experimental realizations. Motivated by transcription networks, we present here a simple way to generate an ensemble of random directed graphs with, asymptotically, scale-free outdegree and compact indegree. Entries in each row of the adjacency matrix are set to be zero or one according to the toss of a biased coin, with a chosen probability distribution for the biases. This defines a quick and simple algorithm, which yields good results already for graphs of size n ~ 100. Perhaps more importantly, many of the relevant observables are accessible analytically, improving upon previous estimates for similar graphs

Gravitational detection of a low-mass dark satellite at cosmological distance

The mass-function of dwarf satellite galaxies that are observed around Local Group galaxies substantially differs from simulations based on cold dark matter: the simulations predict many more dwarf galaxies than are seen. The Local Group, however, may be anomalous in this regard. A massive dark satellite in an early-type lens galaxy at z = 0.222 was recently found using a new method based on gravitational lensing, suggesting that the mass fraction contained in substructure could be higher than is predicted from simulations. The lack of very low mass detections, however, prohibited any constraint on their mass function. Here we report the presence of a 1.9 +/- 0.1 x 10^8 M_sun dark satellite in the Einstein-ring system JVAS B1938+666 at z = 0.881, where M_sun denotes solar mass. This satellite galaxy has a mass similar to the Sagittarius galaxy, which is a satellite of the Milky Way. We determine the logarithmic slope of the mass function for substructure beyond the local Universe to be alpha = 1.1^+0.6_-0.4, with an average mass-fraction of f = 3.3^+3.6_-1.8 %, by combining data on both of these recently discovered galaxies. Our results are consistent with the predictions from cold dark matter simulations at the 95 per cent confidence level, and therefore agree with the view that galaxies formed hierarchically in a Universe composed of cold dark matter.Comment: 25 pages, 7 figures, accepted for publication in Nature (19 January 2012

Differential-Algebraic Integrability Analysis of the Generalized Riemann Type and Korteweg-de Vries Hydrodynamical Equations

A differential-algebraic approach to studying the Lax type integrability of the generalized Riemann type hydrodynamic equations at N = 3; 4 is devised. The approach is also applied to studying the Lax type integrability of the well known Korteweg-de Vries dynamical system.Comment: 11 page

Feedback topology and XOR-dynamics in Boolean networks with varying input structure

We analyse a model of fixed in-degree Random Boolean Networks in which the fraction of input-receiving nodes is controlled by a parameter gamma. We investigate analytically and numerically the dynamics of graphs under a parallel XOR updating scheme. This scheme is interesting because it is accessible analytically and its phenomenology is at the same time under control, and as rich as the one of general Boolean networks. Biologically, it is justified on abstract grounds by the fact that all existing interactions play a dynamical role. We give analytical formulas for the dynamics on general graphs, showing that with a XOR-type evolution rule, dynamic features are direct consequences of the topological feedback structure, in analogy with the role of relevant components in Kauffman networks. Considering graphs with fixed in-degree, we characterize analytically and numerically the feedback regions using graph decimation algorithms (Leaf Removal). With varying gamma, this graph ensemble shows a phase transition that separates a tree-like graph region from one in which feedback components emerge. Networks near the transition point have feedback components made of disjoint loops, in which each node has exactly one incoming and one outgoing link. Using this fact we provide analytical estimates of the maximum period starting from topological considerations

The ACS LCID project. X. The Star Formation History of IC 1613: Revisiting the Over-Cooling Problem

We present an analysis of the star formation history (SFH) of a field near the half light radius in the Local Group dwarf irregular galaxy IC 1613 based on deep Hubble Space Telescope Advanced Camera for Surveys imaging. Our observations reach the oldest main sequence turn-off, allowing a time resolution at the oldest ages of ~1 Gyr. Our analysis shows that the SFH of the observed field in IC 1613 is consistent with being constant over the entire lifetime of the galaxy. These observations rule out an early dominant episode of star formation in IC 1613. We compare the SFH of IC 1613 with expectations from cosmological models. Since most of the mass is in place at early times for low mass halos, a naive expectation is that most of the star formation should have taken place at early times. Models in which star formation follows mass accretion result in too many stars formed early and gas mass fractions which are too low today (the "over-cooling problem"). The depth of the present photometry of IC 1613 shows that, at a resolution of ~1 Gyr, the star formation rate is consistent with being constant, at even the earliest times, which is difficult to achieve in models where star formation follows mass assembly.Comment: 13 pages, 12 figures, accepted for publication in the Ap

Thomas Decomposition of Algebraic and Differential Systems

In this paper we consider disjoint decomposition of algebraic and non-linear partial differential systems of equations and inequations into so-called simple subsystems. We exploit Thomas decomposition ideas and develop them into a new algorithm. For algebraic systems simplicity means triangularity, squarefreeness and non-vanishing initials. For differential systems the algorithm provides not only algebraic simplicity but also involutivity. The algorithm has been implemented in Maple

The ISLAndS project II: The Lifetime Star Formation Histories of Six Andromeda dSphs

The Initial Star formation and Lifetimes of Andromeda Satellites (ISLAndS) project uses Hubble Space Telescope imaging to study a representative sample of six Andromeda dSph satellite companion galaxies. The main goal of the program is to determine whether the star formation histories (SFHs) of the Andromeda dSph satellites demonstrate significant statistical differences from those of the Milky Way, which may be attributable to the different properties of their local environments. Our observations reach the oldest main sequence turn-offs, allowing a time resolution at the oldest ages of ~ 1 Gyr, which is comparable to the best achievable resolution in the MW satellites. We find that the six dSphs present a variety of SFHs that are not strictly correlated with luminosity or present distance from M31. Specifically, we find a significant range in quenching times (lookback times from 9 to 6 Gyr), but with all quenching times more than ~ 6 Gyr ago. In agreement with observations of Milky Way companions of similar mass, there is no evidence of complete quenching of star formation by the cosmic UV background responsible for reionization, but the possibility of a degree of quenching at reionization cannot be ruled out. We do not find significant differences between the SFHs of the three members of the vast, thin plane of satellites and the three off-plane dSphs. The primary difference between the SFHs of the ISLAndS dSphs and Milky Way dSph companions of similar luminosities and host distances is the absence of very late quenching (< 5 Gyr ago) dSphs in the ISLAndS sample. Thus, models that can reproduce satellite populations with and without late quenching satellites will be of extreme interest.Comment: 24 pages, 11 figures, 3 tables, submitted to the Ap

The ISLANDS project I: Andromeda XVI, An Extremely Low Mass Galaxy not Quenched by Reionization

Based on data aquired in 13 orbits of HST time, we present a detailed evolutionary history of the M31 dSph satellite Andromeda XVI, including its life-time star formation history, the spatial distribution of its stellar populations, and the properties of its variable stars. And XVI is characterized by prolonged star formation activity from the oldest epochs until star formation was quenched ~6 Gyr ago, and, notably, only half of the mass in stars of And XVI was in place 10 Gyr ago. And XVI appears to be a low mass galaxy for which the early quenching by either reionization or starburst feedback seems highly unlikely, and thus, is most likely due to an environmental effect (e.g., an interaction), possibly connected to a late infall in the densest regions of the Local Group. Studying the star formation history as a function of galactocentric radius, we detect a mild gradient in the star formation history: the star formation activity between 6 and 8 Gyr ago is significantly stronger in the central regions than in the external regions, although the quenching age appears to be the same, within 1 Gyr. We also report the discovery of 9 RR Lyrae stars, 8 of which belong to And XVI. The RR Lyrae stars allow a new estimate of the distance, (m-M)0= 23.72+/-0.09 mag, which is marginally larger than previous estimates based on the tip of the red giant branch.Comment: Accepted for publication on Ap

Shaping the Phase of a Single Photon

While the phase of a coherent light field can be precisely known, the phase of the individual photons that create this field, considered individually, cannot. Phase changes within single-photon wave packets, however, have observable effects. In fact, actively controlling the phase of individual photons has been identified as a powerful resource for quantum communication protocols. Here we demonstrate the arbitrary phase control of a single photon. The phase modulation is applied without affecting the photon's amplitude profile and is verified via a two-photon quantum interference measurement, which can result in the fermionic spatial behaviour of photon pairs. Combined with previously demonstrated control of a single photon's amplitude, frequency, and polarisation, the fully deterministic phase shaping presented here allows for the complete control of single-photon wave packets.Comment: 4 pages, 4 figure