Irredundant Triangular Decomposition

Triangular decomposition is a classic, widely used and well-developed way to represent algebraic varieties with many applications. In particular, there exist sharp degree bounds for a single triangular set in terms of intrinsic data of the variety it represents, and powerful randomized algorithms for computing triangular decompositions using Hensel lifting in the zero-dimensional case and for irreducible varieties. However, in the general case, most of the algorithms computing triangular decompositions produce embedded components, which makes it impossible to directly apply the intrinsic degree bounds. This, in turn, is an obstacle for efficiently applying Hensel lifting due to the higher degrees of the output polynomials and the lower probability of success. In this paper, we give an algorithm to compute an irredundant triangular decomposition of an arbitrary algebraic set $W$ defined by a set of polynomials in C[x_1, x_2, ..., x_n]. Using this irredundant triangular decomposition, we were able to give intrinsic degree bounds for the polynomials appearing in the triangular sets and apply Hensel lifting techniques. Our decomposition algorithm is randomized, and we analyze the probability of success

Some algebraic properties of differential operators

First, we study the subskewfield of rational pseudodifferential operators over a differential field K generated in the skewfield of pseudodifferential operators over K by the subalgebra of all differential operators. Second, we show that the Dieudonne' determinant of a matrix pseudodifferential operator with coefficients in a differential subring A of K lies in the integral closure of A in K, and we give an example of a 2x2 matrix differential operator with coefficients in A whose Dieudonne' determiant does not lie in A.Comment: 15 page

Tannakian approach to linear differential algebraic groups

Tannaka's Theorem states that a linear algebraic group G is determined by the category of finite dimensional G-modules and the forgetful functor. We extend this result to linear differential algebraic groups by introducing a category corresponding to their representations and show how this category determines such a group.Comment: 31 pages; corrected misprint

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

The Birth and Growth of Neutralino Haloes

We use the Extended-Press-Schechter (EPS) formalism to study halo assembly histories in a standard $\Lambda$CDM cosmology. A large ensemble of Monte Carlo random walks provides the {\it entire} halo membership histories of a representative set of dark matter particles, which we assume to be neutralinos. The first generation halos of most particles do not have a mass similar to the free-streaming cut-off $M_{f.s.}$ of the neutralino power spectrum, nor do they form at high redshift. Median values are $M_1 = 10^5$ to $10^7M_{f.s.}$ and $z_1 = 13$ to 8 depending on the form of the collapse barrier assumed in the EPS model. For almost a third of all particles the first generation halo has $M_1>10^9M_{f.s.}$. At redshifts beyond 20, most neutralinos are not yet part of any halo but are still diffuse. These numbers apply with little modification to the neutralinos which are today part of halos similar to that of the Milky Way. Up to 10% of the particles in such halos were never part of a smaller object; the typical particle has undergone $\sim 5$ "accretion events' where the halo it was part of falls into a more massive object. Available N-body simulations agree well with the EPS predictions for an "ellipsoidal" collapse barrier, so these may provide a reliable extension of simulation results to smaller scales. The late formation times and large masses of the first generation halos of most neutralinos imply that they will be disrupted with high efficiency during halo assembly.Comment: 7 pages, 7 figure

A Bose-Einstein Approach to the Random Partitioning of an Integer

Consider N equally-spaced points on a circle of circumference N. Choose at random n points out of $N$ on this circle and append clockwise an arc of integral length k to each such point. The resulting random set is made of a random number of connected components. Questions such as the evaluation of the probability of random covering and parking configurations, number and length of the gaps are addressed. They are the discrete versions of similar problems raised in the continuum. For each value of k, asymptotic results are presented when n,N both go to infinity according to two different regimes. This model may equivalently be viewed as a random partitioning problem of N items into n recipients. A grand-canonical balls in boxes approach is also supplied, giving some insight into the multiplicities of the box filling amounts or spacings. The latter model is a k-nearest neighbor random graph with N vertices and kn edges. We shall also briefly consider the covering problem in the context of a random graph model with N vertices and n (out-degree 1) edges whose endpoints are no more bound to be neighbors

SubHaloes going Notts: The SubHalo-Finder Comparison Project

We present a detailed comparison of the substructure properties of a single Milky Way sized dark matter halo from the Aquarius suite at five different resolutions, as identified by a variety of different (sub-)halo finders for simulations of cosmic structure formation. These finders span a wide range of techniques and methodologies to extract and quantify substructures within a larger non-homogeneous background density (e.g. a host halo). This includes real-space, phase-space, velocity-space and time- space based finders, as well as finders employing a Voronoi tessellation, friends-of-friends techniques, or refined meshes as the starting point for locating substructure.A common post-processing pipeline was used to uniformly analyse the particle lists provided by each finder. We extract quantitative and comparable measures for the subhaloes, primarily focusing on mass and the peak of the rotation curve for this particular study. We find that all of the finders agree extremely well on the presence and location of substructure and even for properties relating to the inner part part of the subhalo (e.g. the maximum value of the rotation curve). For properties that rely on particles near the outer edge of the subhalo the agreement is at around the 20 per cent level. We find that basic properties (mass, maximum circular velocity) of a subhalo can be reliably recovered if the subhalo contains more than 100 particles although its presence can be reliably inferred for a lower particle number limit of 20. We finally note that the logarithmic slope of the subhalo cumulative number count is remarkably consistent and <1 for all the finders that reached high resolution. If correct, this would indicate that the larger and more massive, respectively, substructures are the most dynamically interesting and that higher levels of the (sub-)subhalo hierarchy become progressively less important.Comment: 16 pages, 7 figures, 2 tables, Accepted for MNRA

The differential-algebraic and bi-Hamiltonian integrability analysis of the Riemann type hierarchy revisited

A differential-algebraic approach to studying the Lax type integrability of the generalized Riemann type hydrodynamic hierarchy is revisited, its new Lax type representation and Poisson structures constructed in exact form. The related bi-Hamiltonian integrability and compatible Poissonian structures of the generalized Riemann type hierarchy are also discussed.Comment: 18 page

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

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