Frattali autosimili e misura di Hausdorff

In questa tesi si vuole trattare il concetto di dimensione a partire dalla teoria della misura e lo si vuole applicare per definire e studiare gli insiemi frattali, in particolare, autosimili. Nel primo capitolo si tratta la dimensione di Hausdorff a partire dalla teoria della misura di Hausdorff, di cui si osservano alcune delle proprietà grazie a cui la prima si può definire. Nel secondo capitolo si danno altre definizioni di dimensione, come ad esempio quella di auto-similarità e la box-counting, per mostrare che tale concetto non è univoco. Si analizzano quindi le principali differenze tra le diverse dimensioni citate e si forniscono esempi di insiemi per cui esse coincidono e altri, invece, per cui esse differiscono. Nel terzo capitolo si introduce poi il vero e proprio concetto di insieme frattale. In particolare, definendo i sistemi di funzioni iterate e studiandone le principali proprietà, si definisce una particolare classe di insiemi frattali detti autosimili. In questo capitolo sono enunciati e dimostrati teoremi fondamentali che legano gli attrattori di sistemi di funzioni iterate a insiemi frattali autosimili e forniscono, per alcuni specifici casi, una formula per calcolarne la dimensione di Hausdorff. Si danno, inoltre, esempi di calcolo di tale dimensione per alcuni insiemi frattali autosimili molto noti. Nel quarto capitolo si dà infine un esempio di funzione che abbia grafico frattale, la Funzione di Weierstrass, per mostrare un caso pratico in cui è utilizzata la teoria studiata nei capitoli precedenti.

Perspectives for Forensic Intelligence in anti-doping: thinking outside the box

Today's approach to anti-doping is mostly centered on the judicial process, despite pursuing a further goal in the detection, reduction, solving and/or prevention of doping. Similarly to decision-making in the area of law enforcement feeding on Forensic Intelligence, anti-doping might significantly benefit from a more extensive gathering of knowledge. Forensic Intelligence might bring a broader logical dimension to the interpretation of data on doping activities for a more future-oriented and comprehensive approach instead of the traditional case-based and reactive process. Information coming from a variety of sources related to doping, whether directly or potentially, would feed an organized memory to provide real time intelligence on the size, seriousness and evolution of the phenomenon. Due to the complexity of doping, integrating analytical chemical results and longitudinal monitoring of biomarkers with physiological, epidemiological, sociological or circumstantial information might provide a logical framework enabling fit for purpose decision-making. Therefore, Anti-Doping Intelligence might prove efficient at providing a more proactive response to any potential or emerging doping phenomenon or to address existing problems with innovative actions or/and policies. This approach might prove useful to detect, neutralize, disrupt and/or prevent organized doping or the trafficking of doping agents, as well as helping to refine the targeting of athletes or teams. In addition, such an intelligence-led methodology would serve to address doping offenses in the absence of adverse analytical chemical evidence.

Perspectives for Forensic Intelligence in anti-doping: Thinking outside of the box.

Today's approach to anti-doping is mostly centered on the judicial process, despite pursuing a further goal in the detection, reduction, solving and/or prevention of doping. Similarly to decision-making in the area of law enforcement feeding on Forensic Intelligence, anti-doping might significantly benefit from a more extensive gathering of knowledge. Forensic Intelligence might bring a broader logical dimension to the interpretation of data on doping activities for a more future-oriented and comprehensive approach instead of the traditional case-based and reactive process. Information coming from a variety of sources related to doping, whether directly or potentially, would feed an organized memory to provide real time intelligence on the size, seriousness and evolution of the phenomenon. Due to the complexity of doping, integrating analytical chemical results and longitudinal monitoring of biomarkers with physiological, epidemiological, sociological or circumstantial information might provide a logical framework enabling fit for purpose decision-making. Therefore, Anti-Doping Intelligence might prove efficient at providing a more proactive response to any potential or emerging doping phenomenon or to address existing problems with innovative actions or/and policies. This approach might prove useful to detect, neutralize, disrupt and/or prevent organized doping or the trafficking of doping agents, as well as helping to refine the targeting of athletes or teams. In addition, such an intelligence-led methodology would serve to address doping offenses in the absence of adverse analytical chemical evidence.

Non-planar double-box, massive and massless pentabox Feynman integrals in the negative-dimensional approach

The negative-dimensional integration method is a technique which can be applied, with success, in usual covariant gauge calculations. We consider three two-loop diagrams: the scalar massless non-planar double-box with six propagators and the scalar pentabox in two cases, where six virtual particles have the same mass, and in the case all of them are massless. Our results are given in terms of hypergeometric functions of Mandelstam variables and also for arbitrary exponents of propagators and dimension D.

Hausdorff dimension for non-hyperbolic repellers II: Da diffeomorphisms

We study non-hyperbolic repellers of diffeomorphisms derived from transitive Anosov diffeomorphisms with unstable dimension 2 through a Hopf bifurcation. Using some recent abstract results about non-uniformly expanding maps with holes, by ourselves and by Dysman, we show that the Hausdorff dimension and the limit capacity (box dimension) of the repeller are strictly less than the dimension of the ambient manifold.

Fractal dimension of the mandibular trabecular bone measured on digital and digitized images.

Aims: This study compared fractal dimension (FD) values on mandibular trabecular bone in digital and digitized images at different spatial and contrast resolutions. Materials and Methods: 12 radiographs of dried human mandibles were obtained using custom-fabricated hybrid image receptors composed of a periapical radiographic film and a photostimulable phosphor plate (PSP). The film/ PSP sets were disassembled, and the PSPs produced images with 600 dots per inch (dpi) and 16 bits. These images were exported as tagged image file format (TIFF), 16 and 8 bits, and 600, 300 and 150 dpi. The films were processed and digitized 3 times on a flatbed scanner, producing TIFF images with 600, 300 and 150 dpi, and 8 bits. On each image, a circular region of interest was selected on the trabecular alveolar bone, away from root apices and FD was calculated by tile counting method. Two-way ANOVA and Tukey’s test were conducted to compare the mean values of FD, according to image type and spatial resolution (α = 5%). Results: Spatial resolution was directly and inversely proportional to FD mean values and standard deviation, respectively. Spatial resolution of 150 dpi yielded significant lower mean values of FD than the resolutions of 600 and 300 dpi ( P < 0.05). A nonsignificant variability was observed for the image types ( P > 0.05). The interaction between type of image and level of spatial resolution was not signi fi cant (P > 0.05). Conclusion: Under the tested, conditions, FD values of the mandibular trabecular bone assessed either by digital or digitized images did not change. Furthermore, these values were in fluenced by lower spatial resolution but not by contrast resolution.

Counting singularities via fitting ideals

The stable singularities of differential map germs constitute the main source of studying the geometric and topological behavior of these maps. In particular, one interesting problem is to find formulae which allow us to count the isolated stable singularities which appear in the discriminant of a stable deformation of a finitely determined map germ. Mond and Pellikaan showed how the Fitting ideals are related to such singularities and obtain a formula to count the number of ordinary triple points in map germs from C-2 to C-3, in terms of the Fitting ideals associated with the discriminant. In this article we consider map germs from (Cn+m, 0) to (C-m, 0), and obtain results to count the number of isolated singularities by means of the dimension of some associated algebras to the Fitting ideals. First in Corollary 4.5 we provide a way to compute the total sum of these singularities. In Proposition 4.9, for m = 3 we show how to compute the number of ordinary triple points. In Corollary 4.10 and with f of co-rank one, we show a way to compute the number of points formed by the intersection between a germ of a cuspidal edge and a germ of a plane. Furthermore, we show in some examples how to calculate the number of isolated singularities using these results.