Symmetrization in Geometry

The concept of an $i$-symmetrization is introduced, which provides a convenient framework for most of the familiar symmetrization processes on convex sets. Various properties of $i$-symmetrizations are introduced and the relations between them investigated. New expressions are provided for the Steiner and Minkowski symmetrals of a compact convex set which exhibit a dual relationship between them. Characterizations of Steiner, Minkowski and central symmetrization, in terms of natural properties that they enjoy, are given and examples are provided to show that none of the assumptions made can be dropped or significantly weakened. Other familiar symmetrizations, such as Schwarz symmetrization, are discussed and several new ones introduced.Comment: A chacterization of central symmetrization has been added and several typos have been corrected. This version has been accepted for publication on Advances in Mathematic

Orbit Determination with the two-body Integrals

We investigate a method to compute a finite set of preliminary orbits for solar system bodies using the first integrals of the Kepler problem. This method is thought for the applications to the modern sets of astrometric observations, where often the information contained in the observations allows only to compute, by interpolation, two angular positions of the observed body and their time derivatives at a given epoch; we call this set of data attributable. Given two attributables of the same body at two different epochs we can use the energy and angular momentum integrals of the two-body problem to write a system of polynomial equations for the topocentric distance and the radial velocity at the two epochs. We define two different algorithms for the computation of the solutions, based on different ways to perform elimination of variables and obtain a univariate polynomial. Moreover we use the redundancy of the data to test the hypothesis that two attributables belong to the same body (linkage problem). It is also possible to compute a covariance matrix, describing the uncertainty of the preliminary orbits which results from the observation error statistics. The performance of this method has been investigated by using a large set of simulated observations of the Pan-STARRS project.Comment: 23 pages, 1 figur

Platonic polyhedra, periodic orbits and chaotic motions in the N-body problem with non-Newtonian forces

We consider the $N$-body problem with interaction potential $U_alpha=rac{1}{ert x_i-x_jert^alpha}$ for alpha>1. We assume that the particles have all the same mass and that $N$ is the order $ertmathcal{R}ert$ of the rotation group $mathcal{R}$ of one of the five Platonic polyhedra. We study motions that, up to a relabeling of the $N$ particles, are invariant under $mathcal{R}$. By variational techniques we prove the existence of periodic and chaotic motions

Long term dynamics for the restricted N-body problem with mean motion resonances and crossing singularities

We consider the long term dynamics of the restricted N -body problem, modeling in a statistical sense the motion of an asteroid in the gravitational field of the Sun and the solar system planets. We deal with the case of a mean motion resonance with one planet and assume that the osculating trajectory of the asteroid crosses the one of some planet, possibly different from the resonant one, during the evolution. Such crossings produce singularities in the differential equations for the motion of the asteroid, obtained by standard perturbation theory. In this work we prove that the vector field of these equations can be extended to two locally Lipschitz-continuous vector fields on both sides of a set of crossing conditions. This allows us to define generalized solutions, continuous but not differentiable, going beyond these singularities. Moreover, we prove that the long term evolution of the ’signed’ orbit distance (Gronchi and Tommei 2007) between the asteroid and the planet is differentiable in a neighborhood of the crossing times. In case of crossings with the resonant planet we recover the known dynamical protection mechanism against collisions. We conclude with a numerical comparison between the long term and the full evolutions in the case of asteroids belonging to the ’Alinda’ and ’Toro’ classes (Milani et al. 1989). This work extends the results in (Gronchi and Tardioli 2013) to the relevant case of asteroids in mean motion resonance with a planet

On the possible values of the orbit distance between a near-Earth asteroid and the Earth

We consider all the possible trajectories of a near-Earth asteroid (NEA), corresponding to the whole set of heliocentric orbital elements with perihelion distance q ≤ 1.3 au and eccentricity e ≤ 1 (NEA class). For these hypothetical trajectories, we study the range of the values of the distance from the trajectory of the Earth (assumed on a circular orbit) as a function of selected orbital elements of the asteroid. The results of this geometric approach are useful to explain some aspects of the orbital distribution of the known NEAs. We also show that the maximal orbit distance between an object in the NEA class and the Earth is attained by a parabolic orbit, with apsidal line orthogonal to the ecliptic plane. It turns out that the threshold value of q for the NEA class (qmax = 1.3 au) is very close to a critical value, below which the above result is not valid

Maps of secular resonances in the NEO region

Context. From numerical simulations, it is known that some secular resonances may affect the motion of near-Earth objects (NEOs). However, the specific location of the secular resonance inside the NEO region is not fully known, because the methods previously used to predict their location can not be used for highly eccentric orbits and when the NEOs cross the orbits of the planets. Aims. In this paper, we aim to map the secular resonances with the planets from Venus to Saturn in the NEO region, even for high values of the eccentricity. Methods. We used an averaged semi-analytical model that can deal with orbit crossing singularities for the computation of the secular dynamics of NEOs, from which we can obtain suitable proper elements and proper frequencies. Then, we computed the proper frequencies over a uniform grid in the proper elements space. Secular resonances are thus located by the level curves corresponding to the proper frequencies of the planets. Results. We determined the location of the secular resonances with the planets from Venus to Saturn, showing that they appear well inside the NEO region. By using full numerical N-body simulations we also showed that the location predicted by our method is fairly accurate. Finally, we provided some indications about possible dynamical paths inside the NEO region, due to the presence of secular resonances.Comment: Accepted for publication in A&

Orbit Determination with the two-body Integrals. II

The first integrals of the Kepler problem are used to compute preliminary orbits starting from two short observed arcs of a celestial body, which may be obtained either by optical or radar observations. We write polynomial equations for this problem, that we can solve using the powerful tools of computational Algebra. An algorithm to decide if the linkage of two short arcs is successful, i.e. if they belong to the same observed body, is proposed and tested numerically. In this paper we continue the research started in [Gronchi, Dimare, Milani, 'Orbit determination with the two-body intergrals', CMDA (2010) 107/3, 299-318], where the angular momentum and the energy integrals were used. A suitable component of the Laplace-Lenz vector in place of the energy turns out to be convenient, in fact the degree of the resulting system is reduced to less than half.Comment: 15 pages, 4 figure

ELECTRO-OSMOTIC DEWATERING OF SEWAGE SLUDGE: PRELIMINARY RESULTS

About half of the organic pollution load treated by the activated sludge process is oxidised and converted into water and carbon dioxide, while the remaining is converted into biomass, called “excess biological sludge” or “waste sludge”. At present, this technique is the cheapest way to remove colloidal and soluble organic pollutants from sewage, but it produces a huge amount of liquid waste sludge, with a dry solid (DS) content of less than 2%, rich in organic substances, mostly biodegradable. Therefore, it needs further processes to reduce its volume, by reducing its water content, and to reduce its polluting potential, due to its high content of biodegradable organic matter. However, sludge produced by Wastewater Treatment Plants (WWTPs) usually has a poor dewaterability. Industrially, mechanical dewatering is the method employed to increase the dry solids (DS) of sewage sludge. The aim of dewatering is the decrease of transport and disposal costs, but the efficiency of the mechanical dehydration methods is low. Electro-osmosis could be a suitable technique to reduce water content of the final sludge by the application of an electric field. Studies about electro-dewatering reported that the DS could increase to values around 40-45%, with respect to 25-30% obtained by mechanical methods, such as centrifuge, filter press and belt press. Aim Here, we investigated the parameters affecting pressure-driven electro-dewatering, with a static and a dynamic device, starting from types of sludge mechanically pre-dehydrated. We focused on the optimization of operating conditions to achieve the best results in terms of solids percentage at the minimum energy consumption. Methods Experiments have been performed by means of a lab-scale device able to produce both a mechanical pressure and an electric field. The use of both these systems could be efficient to increase the amount of dry solids in the final sludge. This device, as illustrated in Figure 1, consists of: • Cylindrical glass vessel (h=176 mm, Ø=80 mm) • Cooling water-jacket • Compressed air system (1-4.5 bar) • Double effect cylinder (200 mm stroke) SMC-CP96 (piston) • DC power supply (30 V-5 A) • Anode: DSA Ti (MMO), fixed to the piston • Cathode: stainless steel mesh (AISI 304) • Cloth: PTT (polytrimethyleneterephthalate) Figure 1-Equipment schematic of electro-dewatering device. Main Results The applied voltage has been set at 10 V, 15 V and 20 V and pressures at 3 bar. The results confirmed values of final DS around 40%. Initial DS and sludge conductivity, together with the applied electric field, are the key factors of electro-osmosis tests. The rotating piston highlighted that a mixing of sludge during the potential application could slightly increase the sludge conductivity, but cakes must be kept under 1.5 cm of thickness in order to achieve a good dewatering. In conclusion, in this paper we assess the feasibility of the electro-osmosis dewatering. It is a promising technique to remove water from sewage sludge, since it is able to increase DS of 15% on average with respect industrial mechanical dewatering. Moreover, this work proved that the treatment of a high DS cake is more difficult than a non-dewatered sludge due to a lower conductivity during the process