A moment symbolic representation of Lévy processes with applications

By using a symbolic method, known in the literature as the classical umbral calculus, a symbolic representation of Lévy processes is given and a new family of time-space harmonic polynomials with respect to such processes, which includes and generalizes the exponential complete Bell polynomials, is introduced. The usefulness of time-space harmonic polynomials with respect to Lévy processes is that it is a martingale the stochastic process obtained by replacing the indeterminate x of the polynomials with a Lévy process, whereas the Lévy process does not necessarily have this property. Therefore to find such polynomials could be particularly meaningful for applications. This new family includes Hermite polynomials, time-space harmonic with respect to Brownian motion, Poisson-Charlier polynomials with respect to Poisson processes, Laguerre and actuarial polynomials with respect to Gamma processes , Meixner polynomials of the first kind with respect to Pascal processes, Euler, Bernoulli, Krawtchuk, and pseudo-Narumi polynomials with respect to suitable random walks. The role played by cumulants is stressed and brought to the light, either in the symbolic representation of Lévy processes and their infinite divisibility property, either in the generalization, via umbral Kailath-Segall formula, of the well-known formulae giving elementary symmetric polynomials in terms of power sum symmetric polynomials. The expression of the family of time-space harmonic polynomials here introduced has some connections with the so-called moment representation of various families of multivariate polynomials. Such moment representation has been studied here for the first time in connection with the time-space harmonic property with respect to suitable symbolic multivariate Lévy processes. In particular, multivariate Hermite polynomials and their properties have been studied in connection with a symbolic version of the multivariate Brownian motion, while multivariate Bernoulli and Euler polynomials are represented as powers of multivariate polynomials which are time-space harmonic with respect to suitable multivariate Lévy processes.

Aspetti innovativi per la valutazione e la misura della porosità nei materiali dell'edilizia antica e moderna. La geometria frattale della porosità

New concepts on porosity appraisal in ancient and modern construction materials. The role of Fractal Geometry on porosity characterization and transport phenomena. This work studied the potential of Fractal Geometry to the characterization of porous materials. Besides the descriptive aspects of the pore size distribution, the fractal dimensions have led to the development of rational relations for the prediction of permeability coefficients to fluid and heat transfer. The research considered natural materials used in historical buildings (rock and earth) as well as currently employed materials as hydraulic cement and technologically advanced materials such as silicon carbide or YSZ ceramics. The experimental results of porosity derived from the techniques of mercury intrusion and from the image analysis. Data elaboration was carried out according to established procedures of Fractal Geometry. It was found that certain classes of materials are clearly fractal and respond to simple patterns such as Sierpinski and Menger models. In several cases, however, the fractal character is not recognised because the microstructure of the material is based on different phases at different dimensional scales, and in consequence the “fractal dimensions” calculated from porosimetric data do not come within the standard range (less than 3). Using different type and numbers of fractal units is possible, however, to obtain “virtual” microstructures that have the fraction of voids and pore size distribution equivalent with the experimental ones for almost any material. Thus it was possible to take the expressions for the permeability and the thermal conduction which does not require empirical “constants”, these expressions have also provided values that are generally in agreement with the experimental available data. More problematic has been the fractal discussion of the geometry of the rupture of the material subjected to mechanical stress both external and internal applied. The results achieved on these issues are qualitative and prone to future studies. Keywords: Materials, Microstructure, Porosity, Fractal Geometry, Permeability, Thermal conduction, Mechanical strength.

Enhancing the efficiency of solar concentrators by controlled optical aberrations

The diameters of traditional dish concentrators can reach several tens of meters, the construction of monolithic mirrors being difficult at these scales: cheap flat reflecting facets mounted on a common frame generally reproduce a paraboloidal surface. When a standard imaging mirror is coupled with a PV dense array, problems arise since the solar image focused is intrinsically circular. Moreover, the corresponding irradiance distribution is bell-shaped in contrast with the requirement of having all the cells under the same illumination. Mismatch losses occur when interconnected cells experience different conditions, in particular in series connections. In this PhD Thesis, we aim at solving these issues by a multidisciplinary approach, exploiting optical concepts and applications developed specifically for astronomical use, where the improvement of the image quality is a very important issue. The strategy we propose is to boost the spot uniformity acting uniquely on the primary reflector and avoiding the big mirrors segmentation into numerous smaller elements that need to be accurately mounted and aligned. In the proposed method, the shape of the mirrors is analytically described by the Zernike polynomials and its optimization is numerically obtained to give a non-imaging optics able to produce a quasi-square spot, spatially uniform and with prescribed concentration level. The freeform primary optics leads to a substantial gain in efficiency without secondary optics. Simple electrical schemes for the receiver are also required. The concept has been investigated theoretically modeling an example of CPV dense array application, including the development of non-optical aspects as the design of the detector and of the supporting mechanics. For the method proposed and the specific CPV system described, a patent application has been filed in Italy with the number TO2014A000016. The patent has been developed thanks to the collaboration between the University of Bologna and INAF (National Institute for Astrophysics).

The spatial association of numbers and mathematical symbols

The "SNARC effect" refers to the finding that people respond faster to small numbers with the left hand and to large numbers with the right hand. This effect is often explained by hypothesizing that numbers are represented from left to right in ascending order (Mental Number Line). However, the SNARC effect may not depend on quantitative information, but on other factors such as the order in which numbers are often represented from left to right in our culture. Four experiments were performed to test this hypothesis. In the first experiment, the concept of spatial association was extended to nonnumeric mathematical symbols: the minus and plus symbols. These symbols were presented as fixation points in a spatial compatibility paradigm. The results demonstrated an opposite influence of the two symbols on the target stimulus: the minus symbol tends to favor the target presented on the left, while the plus symbol the target presented on the right, demonstrating that spatial association can emerge in the absence of a numerical context. In the last three experiments, the relationship between quantity and order was evaluated using normal numbers and mirror numbers. Although mirror numbers denote quantity, they are not encountered in a left-to-right spatial organization. In Experiments 1 and 2, participants performed a magnitude classification task with mirror and normal numbers presented together (Experiment 1) or separately (Experiment 2). In Experiment 3, participants performed a new task in which quantity information processing was not required: the mirror judgment task. The results show that participants access the quantity of both normal and mirror numbers, but only the normal numbers are spatially organized from left to right. In addition, the physical similarity between the numbers, used as a predictor variable in the last three experiments, showed that the physical characteristics of numbers influenced participants' reaction times.