A combinatorial description of finite O-sequences and aCM genera

The goal of this paper is to explicitly detect all the arithmetic genera of arithmetically Cohen-Macaulay projective curves with a given degree $d$. It is well-known that the arithmetic genus $g$ of a curve $C$ can be easily deduced from the $h$-vector of the curve; in the case where $C$ is arithmetically Cohen-Macaulay of degree $d$, $g$ must belong to the range of integers $\big\{0,\ldots,\binom{d-1}{2}\big\}$. We develop an algorithmic procedure that allows one to avoid constructing most of the possible $h$-vectors of $C$. The essential tools are a combinatorial description of the finite O-sequences of multiplicity $d$, and a sort of continuity result regarding the generation of the genera. The efficiency of our method is supported by computational evidence. As a consequence, we single out the minimal possible Castelnuovo-Mumford regularity of a curve with Cohen-Macaulay postulation and given degree and genus.Comment: Final versio

Segments and Hilbert schemes of points

Using results obtained from the study of homogeneous ideals sharing the same initial ideal with respect to some term order, we prove the singularity of the point corresponding to a segment ideal with respect to the revlex term order in the Hilbert scheme of points in $\mathbb{P}^n$. In this context, we look inside properties of several types of "segment" ideals that we define and compare. This study led us to focus our attention also to connections between the shape of generators of Borel ideals and the related Hilbert polynomial, providing an algorithm for computing all saturated Borel ideals with the given Hilbert polynomial.Comment: 19 pages, 2 figures. Comments and suggestions are welcome

Photon elastic scattering simulation: validation and improvements to Geant4

Several models for the simulation of photon elastic scattering are quantitatively evaluated with respect to a large collection of experimental data retrieved from the literature. They include models based on the form factor approximation, on S-matrix calculations and on analytical parameterizations; they exploit publicly available data libraries and tabulations of theoretical calculations. Some of these models are currently implemented in general purpose Monte Carlo systems; some have been implemented and evaluated for the first time in this paper for possible use in Monte Carlo particle transport. The analysis mainly concerns the energy range between 5 keV and a few MeV. The validation process identifies the newly implemented model based on second order S-matrix calculations as the one best reproducing experimental measurements. The validation results show that, along with Rayleigh scattering, additional processes, not yet implemented in Geant4 nor in other major Monte Carlo systems, should be taken into account to realistically describe photon elastic scattering with matter above 1 MeV. Evaluations of the computational performance of the various simulation algorithms are reported along with the analysis of their physics capabilities

Minimal Castelnuovo-Mumford regularity for a given Hilbert polynomial

Let $K$ be an algebraically closed field of null characteristic and $p(z)$ a Hilbert polynomial. We look for the minimal Castelnuovo-Mumford regularity $m_{p(z)}$ of closed subschemes of projective spaces over $K$ with Hilbert polynomial $p(z)$. Experimental evidences led us to consider the idea that $m_{p(z)}$ could be achieved by schemes having a suitable minimal Hilbert function. We give a constructive proof of this fact. Moreover, we are able to compute the minimal Castelnuovo-Mumford regularity $m_p(z)^{\varrho}$ of schemes with Hilbert polynomial $p(z)$ and given regularity $\varrho$ of the Hilbert function, and also the minimal Castelnuovo-Mumford regularity $m_u$ of schemes with Hilbert function $u$. These results find applications in the study of Hilbert schemes. They are obtained by means of minimal Hilbert functions and of two new constructive methods which are based on the notion of growth-height-lexicographic Borel set and called ideal graft and extended lifting.Comment: 21 pages. Comments are welcome. More concise version with a slight change in the title. A further revised version has been accepted for publication in Experimental Mathematic

Ionization cross sections for low energy electron transport

Two models for the calculation of ionization cross sections by electron impact on atoms, the Binary-Encouter-Bethe and the Deutsch-Maerk models, have been implemented; they are intended to extend and improve Geant4 simulation capabilities in the energy range below 1 keV. The physics features of the implementation of the models are described, and their differences with respect to the original formulations are discussed. Results of the verification with respect to the original theoretical sources and of extensive validation with respect to experimental data are reported. The validation process also concerns the ionization cross sections included in the Evaluated Electron Data Library used by Geant4 for low energy electron transport. Among the three cross section options, the Deutsch-Maerk model is identified as the most accurate at reproducing experimental data over the energy range subject to test.Comment: To be published in IEEE Trans. Nucl. Sci., Dec. 201

Physics-related epistemic uncertainties in proton depth dose simulation

A set of physics models and parameters pertaining to the simulation of proton energy deposition in matter are evaluated in the energy range up to approximately 65 MeV, based on their implementations in the Geant4 toolkit. The analysis assesses several features of the models and the impact of their associated epistemic uncertainties, i.e. uncertainties due to lack of knowledge, on the simulation results. Possible systematic effects deriving from uncertainties of this kind are highlighted; their relevance in relation to the application environment and different experimental requirements are discussed, with emphasis on the simulation of radiotherapy set-ups. By documenting quantitatively the features of a wide set of simulation models and the related intrinsic uncertainties affecting the simulation results, this analysis provides guidance regarding the use of the concerned simulation tools in experimental applications; it also provides indications for further experimental measurements addressing the sources of such uncertainties.Comment: To be published in IEEE Trans. Nucl. Sc

Auroral Radio Emission from Stars: the case of CU Virginis

CU Virginis is a rapidly rotating Magnetic Chemically Peculiar star with at present unique characteristics as radio emitter. The most intriguing one is the presence of intense, 100% circularly polarized radiation ascribed to Cyclotron Maser. Each time the star rotates, this highly beamed emission points two times toward the Earth, like a pulsar. We observed CU Vir in April 2010 with the EVLA in two bands centered at 1450 and 1850 MHz. We covered nearly the whole rotational period, confirming the presence of the two pulses at a flux density up to 20 mJy. Dynamical spectra, obtained with unprecedented spectral and temporal sensitivity, allow us to clearly see the different time delays as a function of the frequency. We interpret this behaviour as a propagation effect of the radiation inside the stellar magnetosphere. The emerging scenario suggests interesting similarities with the auroral radio emission from planets, in particular with the Auroral Kilometric Radiation (AKR) from Earth, which originates at few terrestrial radii above the magnetic poles and was only recently discovered to be highly beamed. We conclude that the magnetospheres of CU Vir, Earth and other planets, maybe also exoplanets, could have similar geometrical and physical characteristics in the regions where the cyclotron maser is generated. In addition, the pulses are perfect "markers" of the rotation period. This has given us for the first time the possibility to measure with extraordinary accuracy the spin down of a star on or near the main sequence.Comment: 18 pages, 4 figures, Accepted to APJ Letter, EVLA special issu