Automorphisms of the fine grading of sl(n,C) associated with the generalized Pauli matrices

We consider the grading of $sl(n,\mathbb{C})$ by the group $\Pi_n$ of generalized Pauli matrices. The grading decomposes the Lie algebra into $n^2-1$ one--dimensional subspaces. In the article we demonstrate that the normalizer of grading decomposition of $sl(n,\mathbb{C})$ in $\Pi_n$ is the group $SL(2, \mathbb{Z}_n)$, where $\mathbb{Z}_n$ is the cyclic group of order $n$. As an example we consider $sl(3,\mathbb{C})$ graded by $\Pi_3$ and all contractions preserving that grading. We show that the set of 48 quadratic equations for grading parameters splits into just two orbits of the normalizer of the grading in $\Pi_3$

Quantum temporal imaging: application of a time lens to quantum optics

We consider application of a temporal imaging system, based on the sum-frequency generation, to a nonclassical, in particular, squeezed optical temporal waveform. We analyze the restrictions on the pump and the phase matching condition in the summing crystal, necessary for preserving the quantum features of the initial waveform. We show that modification of the notion of the field of view in the quantum case is necessary, and that the quantum field of view is much narrower than the classical one for the same temporal imaging system. These results are important for temporal stretching and compressing of squeezed fields, used in quantum-enhanced metrology and quantum communications.Comment: 9 pages, 3 figure

Centralizers of maximal regular subgroups in simple Lie groups and relative congruence classes of representations

In the paper we present a new, uniform and comprehensive description of centralizers of the maximal regular subgroups in compact simple Lie groups of all types and ranks. The centralizer is either a direct product of finite cyclic groups, a continuous group of rank 1, or a product, not necessarily direct, of a continuous group of rank 1 with a finite cyclic group. Explicit formulas for the action of such centralizers on irreducible representations of the simple Lie algebras are given.Comment: 27 page

Spatial Uncertainty of Target Patterns Generated by Different Prediction Models of Landslide Susceptibility

This contribution exposes the relative uncertainties associated with prediction patterns of landslide susceptibility. The patterns are based on relationships between direct and indirect spatial evidence of landslide occurrences. In a spatial database constructed for the modeling, direct evidence is the presence of landslide trigger areas, while indirect evidence is the presence of corresponding multivariate context in the form of digital maps. Five mathematical modeling functions are applied to capture and integrate evidence, indirect and direct, for separating landslide-presence areas from the areas of landslide assumed absence. Empirical likelihood ratios are used first to represent the spatial relationships. These are then combined by the models into prediction scores, ordered, equal-area ranked, displayed, and synthesized as prediction-rate curves. A critical task is assessing how uncertainty levels vary across the different prediction patterns, i.e., the modeling results visualized as fixed, colored groups of ranks. This is obtained by a strategy of iterative cross validation that uses only part of the direct evidence to model the pattern and the rest to validate it as a predictor. The conducted experiments in a mountainous area in northern Italy point at a research challenge that can now be confronted with relative rank-based statistics and iterative cross-validation processes. The uncertainty properties of prediction patterns are mostly unknown nevertheless they are critical for interpreting and justifying prediction results

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

Monte Carlo Study of the Arrival Time Distribution of Particles in Extensive Air Showers in the Energy Range 1-100 TeV

A detailed simulation of vertical showers in atmosphere produced by primary gammas and protons, in the energy range 1-100 TeV, has been performed by means of the FLUKA Monte Carlo code, with the aim of studying the time structure of the shower front at different detector heights. It turns out that the time delay distribution can be fitted using few parameters coincident with the distribution central moments. Such parameters exhibit a smooth behaviour as a function of energy. These results can be used both for detector design and for the interpretation of the existing measurements. Differences in the time structure between gamma and proton induced showers are found and explained in terms of the non-relativistic component of extensive air showers

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