Discrete group transforms on SU(2) X SU(2) and SU(3)

Discrete group transforms on SU(2) X SU(2) and SU(3

A fast - Monte Carlo toolkit on GPU for treatment plan dose recalculation in proton therapy

In the context of the particle therapy a crucial role is played by Treatment Planning Systems (TPSs), tools aimed to compute and optimize the tratment plan. Nowadays one of the major issues related to the TPS in particle therapy is the large CPU time needed. We developed a software toolkit (FRED) for reducing dose recalculation time by exploiting Graphics Processing Units (GPU) hardware. Thanks to their high parallelization capability, GPUs significantly reduce the computation time, up to factor 100 respect to a standard CPU running software. The transport of proton beams in the patient is accurately described through Monte Carlo methods. Physical processes reproduced are: Multiple Coulomb Scattering, energy straggling and nuclear interactions of protons with the main nuclei composing the biological tissues. FRED toolkit does not rely on the water equivalent translation of tissues, but exploits the Computed Tomography anatomical information by reconstructing and simulating the atomic composition of each crossed tissue. FRED can be used as an efficient tool for dose recalculation, on the day of the treatment. In fact it can provide in about one minute on standard hardware the dose map obtained combining the treatment plan, earlier computed by the TPS, and the current patient anatomic arrangement

Four types of special functions of G_2 and their discretization

Properties of four infinite families of special functions of two real variables, based on the compact simple Lie group G2, are compared and described. Two of the four families (called here C- and S-functions) are well known, whereas the other two (S^L- and S^S-functions) are not found elsewhere in the literature. It is shown explicitly that all four families have similar properties. In particular, they are orthogonal when integrated over a finite region F of the Euclidean space, and they are discretely orthogonal when their values, sampled at the lattice points F_M \subset F, are added up with a weight function appropriate for each family. Products of ten types among the four families of functions, namely CC, CS, SS, SS^L, CS^S, SS^L, SS^S, S^SS^S, S^LS^S and S^LS^L, are completely decomposable into the finite sum of the functions. Uncommon arithmetic properties of the functions are pointed out and questions about numerous other properties are brought forward.Comment: 18 pages, 4 figures, 4 table

Virtual Reality Laboratories in Engineering Blended Learning Environments: Challenges and Opportunities

A great number of educational institutions worldwide have had their activities partially or fully interrupted following the outbreak of the COVID-19 pandemic. Consequently, universities have had to take the necessary steps in order to adapt their teaching, including laboratory workshops, to a fully online or mixed mode of delivery while maintaining their academic standards and providing a high-quality student experience. This transition has required, among other efforts, adequate investments in tools, accessibility, content development, and competences as well as appropriate training for both the teaching and administrative staff. In such a complex scenario, Virtual Reality Laboratories (VRLabs), which in the past already proved themselves to be efficient tools supporting the traditional practical activities, could well represent a valid alternative in the hybrid didactic mode of the contemporary educational landscape, rethinking the educational proposal in light of the indications coming from the scientific literature in the pedagogical field. In this context, the present work carries out a critical review of the existent virtual labs developed in the Engineering departments in the last ten years (2010-2020) and includes a pre-pandemic experience of a VRLab tool-StreamFlowVR-within the Hydraulics course of Basilicata University, Italy. This analysis is aimed at highlighting how ready VRLabs are to be exploited not only in emergency but also in ordinary situations, together with valorising an interdisciplinary dialogue between the pedagogical and technological viewpoints, in order to progressively foster a high-quality and evidence-based educational experience

Distributions of secondary muons at sea level from cosmic gamma rays below 10 TeV

The FLUKA Monte Carlo program is used to predict the distributions of the muons which originate from primary cosmic gamma rays and reach sea level. The main result is the angular distribution of muons produced by vertical gamma rays which is necessary to predict the inherent angular resolution of any instrument utilizing muons to infer properties of gamma ray primaries. Furthermore, various physical effects are discussed which affect these distributions in differing proportions.Comment: 36 pages, 13 figures, minor revision, new layou

Affine extension of noncrystallographic Coxeter groups and quasicrystals

Unique affine extensions H^{\aff}_2, H^{\aff}_3 and H^{\aff}_4 are determined for the noncrystallographic Coxeter groups $H_2$, $H_3$ and $H_4$. They are used for the construction of new mathematical models for quasicrystal fragments with 10-fold symmetry. The case of H^{\aff}_2 corresponding to planar point sets is discussed in detail. In contrast to the cut-and-project scheme we obtain by construction finite point sets, which grow with a model specific growth parameter.Comment: (27 pages, to appear in J. Phys. A

The rings of n-dimensional polytopes

Points of an orbit of a finite Coxeter group G, generated by n reflections starting from a single seed point, are considered as vertices of a polytope (G-polytope) centered at the origin of a real n-dimensional Euclidean space. A general efficient method is recalled for the geometric description of G- polytopes, their faces of all dimensions and their adjacencies. Products and symmetrized powers of G-polytopes are introduced and their decomposition into the sums of G-polytopes is described. Several invariants of G-polytopes are found, namely the analogs of Dynkin indices of degrees 2 and 4, anomaly numbers and congruence classes of the polytopes. The definitions apply to crystallographic and non-crystallographic Coxeter groups. Examples and applications are shown.Comment: 24 page

Three dimensional C-, S- and E-transforms

Three dimensional continuous and discrete Fourier-like transforms, based on the three simple and four semisimple compact Lie groups of rank 3, are presented. For each simple Lie group, there are three families of special functions ($C$-, $S$-, and $E$-functions) on which the transforms are built. Pertinent properties of the functions are described in detail, such as their orthogonality within each family, when integrated over a finite region $F$ of the 3-dimensional Euclidean space (continuous orthogonality), as well as when summed up over a lattice grid $F_M\subset F$ (discrete orthogonality). The positive integer $M$ sets up the density of the lattice containing $F_M$. The expansion of functions given either on $F$ or on $F_M$ is the paper's main focus.Comment: 24 pages, 13 figure