An equivariant isomorphism theorem for mod p\mathfrak p reductions of arboreal Galois representations

Let $\phi$ be a quadratic, monic polynomial with coefficients in $\mathcal O_{F,D}[t]$, where $\mathcal O_{F,D}$ is a localization of a number ring $\mathcal O_F$. In this paper, we first prove that if $\phi$ is non-square and non-isotrivial, then there exists an absolute, effective constant $N_\phi$ with the following property: for all primes $\mathfrak p\subseteq\mathcal O_{F,D}$ such that the reduced polynomial $\phi_\mathfrak p\in (\mathcal O_{F,D}/\mathfrak p)[t][x]$ is non-square and non-isotrivial, the squarefree Zsigmondy set of $\phi_{\mathfrak p}$ is bounded by $N_\phi$. Using this result, we prove that if $\phi$ is non-isotrivial and geometrically stable then outside a finite, effective set of primes of $\mathcal O_{F,D}$ the geometric part of the arboreal representation of $\phi_{\mathfrak p}$ is isomorphic to that of $\phi$. As an application of our results we prove R. Jones' conjecture on the arboreal Galois representation attached to the polynomial $x^2+t$.Comment: Comments are welcome

On Mertens-Ces\`aro Theorem for Number Fields

Let $K$ be a number field with ring of integers $\mathcal O$. After introducing a suitable notion of density for subsets of $\mathcal O$, generalizing that of natural density for subsets of $\mathbb Z$, we show that the density of the set of coprime $m$-tuples of algebraic integers is ${1/\zeta_K(m)}$, where $\zeta_K$ is the Dedekind zeta function of $K$.Comment: To appear in the Bulletin of the Australian Mathematical Societ

Irreducible compositions of degree two polynomials over finite fields have regular structure

Let $q$ be an odd prime power and $D$ be the set of monic irreducible polynomials in $\mathbb F_q[x]$ which can be written as a composition of monic degree two polynomials. In this paper we prove that $D$ has a natural regular structure by showing that there exists a finite automaton having $D$ as accepted language. Our method is constructive.Comment: To appear in The Quarterly Journal of Mathematic

Exceptional scatteredness in prime degree

Let $q$ be an odd prime power and $n$ be a positive integer. Let $\ell\in \mathbb F_{q^n}[x]$ be a $q$-linearised $t$-scattered polynomial of linearized degree $r$. Let $d=\max\{t,r\}$ be an odd prime number. In this paper we show that under these assumptions it follows that $\ell=x$. Our technique involves a Galois theoretical characterization of $t$-scattered polynomials combined with the classification of transitive subgroups of the general linear group over the finite field $\mathbb F_q$

Temporal Planning with Intermediate Conditions and Effects

Automated temporal planning is the technology of choice when controlling systems that can execute more actions in parallel and when temporal constraints, such as deadlines, are needed in the model. One limitation of several action-based planning systems is that actions are modeled as intervals having conditions and effects only at the extremes and as invariants, but no conditions nor effects can be specified at arbitrary points or sub-intervals. In this paper, we address this limitation by providing an effective heuristic-search technique for temporal planning, allowing the definition of actions with conditions and effects at any arbitrary time within the action duration. We experimentally demonstrate that our approach is far better than standard encodings in PDDL 2.1 and is competitive with other approaches that can (directly or indirectly) represent intermediate action conditions or effects

Universal rates for reactive ultracold polar molecules in reduced dimensions

Analytic expressions describe universal elastic and reactive rates of quasi-two-dimensional and quasi-one-dimensional collisions of highly reactive ultracold molecules interacting by a van der Waals potential. Exact and approximate calculations for the example species of KRb show that stability and evaporative cooling can be realized for spin-polarized fermions at moderate dipole and trapping strength, whereas bosons or unlike fermions require significantly higher dipole or trapping strengths.Comment: 4 pages, 3 figure

Number Theoretical Locally Recoverable Codes

In this paper we give constructions for infinite sequences of finite non-linear locally recoverable codes $\mathcal C\subseteq \prod\limits^N_{i=1}\mathbb F_{q_i}$ over a product of finite fields arising from basis expansions in algebraic number fields. The codes in our sequences have increasing length and size, constant rate, fixed locality, and minimum distance going to infinity

CVD nano-coating of carbon composites for space materials atomic oxygen shielding

The present work analyzes the possibility to employ carbon nanostructures as a basic material to prevent the erosion effects of atomic oxygen suffered by the carbon fiber reinforced polymeric material used in low earth orbit space environment. The application of thin protecting coatings to base materials is a widely used method for preventing the atomic oxygen induced erosion, and thus degradation. The generic purpose is to integrate carbon nanostructures onto carbon composites surface in order to develop the basic substrate of advanced nanocomposite for atomic oxygen protection. The final goal is the characterization of carbon nanostructures-reinforced carbon composites by means of on-ground atomic oxygen simulation facility, with the future objective to assess and optimize the process of carbon-multiscale advanced composites production. With such an aim, a wide investigation on the methane chemical vapor deposition (CVD) over catalyzed carbon fiber-based substrates has been carried out. The as grown nanostructures have been analyzed in terms of morphology, as well as regarding the main features of the resulting growth (yield, purity, homogeneity, coating uniformity, etc.) and the influence of the deposition route operating parameters (catalyst typology, gas flowing rate, growth time/temperature, etc.). A high degree of reproducibility in terms of the relationship between the carbon deposit type/yield and the main process variables (catalyst and protocol) has been thus obtained. Finally, atomic oxygen ground tests have been conducted in order to evaluate the coating process effectiveness. The on-ground test in atomic oxygen environment, with respect to the performances of the reference carbon composites (in terms of total mass loss and atomic oxygen rate of erosion), showed a worsening for the disordered carbon deposit, while an intriguing improvement was achieved by the high-yield carbon nano-filaments deposition

A new advanced railgun system for debris impact study

The growing quantity of debris in Earth orbit poses a danger to users of the orbital environment, such as spacecraft. It also increases the risk that humans or manmade structures could be impacted when objects reenter Earth's atmosphere. During the design of a spacecraft, a requirement may be specified for the surviv-ability of the spacecraft against Meteoroid / Orbital Debris (M/OD) impacts throughout the mission; further-more, the structure of a spacecraft is designed to insure its integrity during the launch and, if it is reusable, during descent, re-entry and landing. In addition, the structure has to provide required stiffness in order to allow for exact positioning of experiments and antennas, and it has to protect the payload against the space environment. In order to decrease the probability of spacecraft failure caused by M/OD, space maneuver is needed to avoid M/OD if the M/OD has dimensions larger than 10cm, but for M/OD with dimensions less than 1cm M/OD shields are needed for spacecrafts. It is therefore necessary to determine the impact-related failure mechanisms and associated ballistic limit equations (BLEs) for typical spacecraft components and subsys-tems. The methods that are used to obtain the ballistic limit equations are numerical simulations and la-borato-ry experiments. In order to perform an high energy ballistic characterization of layered structures, a new ad-vanced electromagnetic accelerator, called railgun, has been assembled and tuned. A railgun is an electrically powered electromagnetic projectile launcher. Such device is made up of a pair of parallel conducting rails, which a sliding metallic armature is accelerated along by the electromagnetic effect (Lorentz force) of a cur-rent that flows down one rail, into the armature and then back along the other rail, thanks to a high power pulse given by a bank of capacitors. A tunable power supplier is used to set the capacitors charging voltage at the desired level: in this way the Rail Gun energy can be tuned as a function of the desired bullet velocity. This facility is able to analyze both low and high velocity impacts. A numerical simulation is also performed by using the Ansys Autodyn code in order to analyze the damage. The experimental results and numerical simulations show that the railgun-device is a good candidate to perform impact testing of materials in the space debris energy range