A Geant4 simulation code for simulating optical photons in SPECT scintillation detectors

Geant4 is an object oriented toolkit created for the simulation of High-Energy Physics detectors. Geant4 allows an accurate modeling of radiation sources and detector devices, with easy configuration and friendly interface and at the same time with great accuracy in the simulation of physical processes. While most Monte Carlo codes do not allow the simulation of the transport and boundary characteristics for optical photons transport generated by scintillating crystal, Geant4 allows the simulation of the optical photons. In this paper we present an application of the Geant4 program for simulating optical photons in SPECT cameras. We aim to study the light transport within scintillators, photomultiplier tubes and coupling devices. To this end, we simulated a detector based on a scintillator, coupled to a photomultiplier tube through a glass window. We compared simulated results with experimental data and theoretical models, in order to verify the good matching with our simulations. We simulated a pencil beam of 140 keV photons impinging the crystal at different locations. For each condition, we calculated the value of the Pulse Height Centroid and the spread of the charge distribution, as read out by the anode array of the photomultiplier. Finally, the spatial and the energy resolutions of the camera have been estimated by simulated data. In all cases, we found that simulations agree very well with experimental data

Generalized Jacobi identities and ball-box theorem for horizontally regular vector fields

We consider a family of vector fields and we assume a horizontal regularity on their derivatives. We discuss the notion of commutator showing that different definitions agree. We apply our results to the proof of a ball-box theorem and Poincar\'e inequality for nonsmooth H\"ormander vector fields.Comment: arXiv admin note: material from arXiv:1106.2410v1, now three separate articles arXiv:1106.2410v2, arXiv:1201.5228, arXiv:1201.520

The role of fundamental solution in Potential and Regularity Theory for subelliptic PDE

In this survey we consider a general Hormander type operator, represented as a sum of squares of vector fields plus a drift and we outline the central role of the fundamental solution in developing Potential and Regularity Theory for solutions of related PDEs. After recalling the Gaussian behavior at infinity of the kernel, we show some mean value formulas on the level sets of the fundamental solution, which are the starting point to obtain a comprehensive parallel of the classical Potential Theory. Then we show that a precise knowledge of the fundamental solution leads to global regularity results, namely estimates at the boundary or on the whole space. Finally in the problem of regularity of non linear differential equations we need an ad hoc modification of the parametrix method, based on the properties of the fundamental solution of an approximating problem

BV functions and sets of finite perimeters in sub-Riemannian manifolds

We give a notion of BV function on an oriented manifold where a volume form and a family of lower semicontinuous quadratic forms are given. When we consider sub-Riemannian manifolds, our definition coincides with the one given in the more general context of metric measure spaces which are doubling and support a Poincaré inequality. We focus on finite perimeter sets, i.e., sets whose characteristic function is BV, in sub-Riemannian manifolds. Under an assumption on the nilpotent approximation, we prove a blowup theorem, generalizing the one obtained for step-2 Carnot groups

Wick calculus for the square of a Gaussian random variable with application to Young and hypercontractive inequalities

We investigate a probabilistic interpretation of the Wick product associated to the chisquare distribution in the spirit of the results obtained in Ref. 7 for the Gaussian measure. Our main theorem points out a profound difference from the previously studied Gaussian7 and Poissonian12 cases. As an application, we obtain a Young-type inequality for the Wick product associated to the chi-square distribution which contains as a particular case a known Nelson-type hypercontractivity theorem

On a connection between the Poissonian Wick product and the discrete convolution

Inspired by Lemma 3.1 in [4], where a connection between the Gaussian Wick product and the classic convolution product is shown, we prove that the Wick product associated to the Poisson distribution is related to the discrete convolution and hence to the law of the sum of discrete independent random variables. The proof of the main result is based on elementary probabilistic tools and on the properties of the Poisson-Charlier polynomials