Problema omogeneo di Sturm-Liouville

Lo scopo dell’elaborato è dare un'esposizione del problema omogeneo di Sturm-Liouville, ossia lo studio di un tipo particolare di equazioni differenziali ordinarie del secondo ordine soggette a condizioni al contorno.

Il teorema della mappa di Riemann

Il teorema della mappa di Riemann è un risultato fondamentale dell'analisi complessa che afferma l'esistenza di un biolomorfismo tra un qualsiasi dominio semplicemente connesso incluso strettamente nel piano ed il disco unità. Si tratta di un teorema di grande importanza e generalità, dato che non si fa alcuna ipotesi sul bordo del dominio considerato. Inoltre ha applicazioni in diverse aree della matematica, ad esempio nella topologia: può infatti essere usato per dimostrare che due domini semplicemente connessi del piano sono tra loro omeomorfi. Presentiamo in questa tesi due diverse dimostrazioni del teorema.

Il teorema di Riemann-Roch

Superfici di Riemann compatte, divisori, Teorema di Riemann Roch, immersioni nello spazio proiettivo.

Analysis of the Kohn Laplacian on the Heisenberg Group and on Cauchy-Riemann Manifolds

The purpose of this study is to analyse the regularity of a differential operator, the Kohn Laplacian, in two settings: the Heisenberg group and the strongly pseudoconvex CR manifolds. The Heisenberg group is defined as a space of dimension 2n+1 with a product. It can be seen in two different ways: as a Lie group and as the boundary of the Siegel UpperHalf Space. On the Heisenberg group there exists the tangential CR complex. From this we define its adjoint and the Kohn-Laplacian. Then we obtain estimates for the Kohn-Laplacian and find its solvability and hypoellipticity. For stating L^p and Holder estimates, we talk about homogeneous distributions. In the second part we start working with a manifold M of real dimension 2n+1. We say that M is a CR manifold if some properties are satisfied. More, we say that a CR manifold M is strongly pseudoconvex if the Levi form defined on M is positive defined. Since we will show that the Heisenberg group is a model for the strongly pseudo-convex CR manifolds, we look for an osculating Heisenberg structure in a neighborhood of a point in M, and we want this structure to change smoothly from a point to another. For that, we define Normal Coordinates and we study their properties. We also examinate different Normal Coordinates in the case of a real hypersurface with an induced CR structure. Finally, we define again the CR complex, its adjoint and the Laplacian operator on M. We study these new operators showing subelliptic estimates. For that, we don't need M to be pseudo-complex but we ask less, that is, the Z(q) and the Y(q) conditions. This provides local regularity theorems for Laplacian and show its hypoellipticity on M.

Eulero, Mascheroni e Riemann costanti ed ipotesi

In questa tesi si descrivono la funzione zeta di Riemann, la costante di Eulero-Mascheroni e la funzione gamma di Eulero. Si riportano i legami tra questi e si illustra brevemente l'ipotesi di Riemann degli zeri non banali della funzione zeta, ovvero l'ipotesi della distribuzione dei numeri primi nella retta dei numeri reali.

Alternative derivative expansion in Functional RG and application

We give a brief review of the Functional Renormalization method in quantum field theory, which is intrinsically non perturbative, in terms of both the Polchinski equation for the Wilsonian action and the Wetterich equation for the generator of the proper verteces. For the latter case we show a simple application for a theory with one real scalar field within the LPA and LPA' approximations. For the first case, instead, we give a covariant "Hamiltonian" version of the Polchinski equation which consists in doing a Legendre transform of the flow for the corresponding effective Lagrangian replacing arbitrary high order derivative of fields with momenta fields. This approach is suitable for studying new truncations in the derivative expansion. We apply this formulation for a theory with one real scalar field and, as a novel result, derive the flow equations for a theory with N real scalar fields with the O(N) internal symmetry. Within this new approach we analyze numerically the scaling solutions for N=1 in d=3 (critical Ising model), at the leading order in the derivative expansion with an infinite number of couplings, encoded in two functions V(phi) and Z(phi), obtaining an estimate for the quantum anomalous dimension with a 10% accuracy (confronting with Monte Carlo results).

NCX-1000, a nitric oxide-releasing derivative of UDCA, does not decrease portal pressure in patients with cirrhosis: results of a randomized, double-blind, dose-escalating study

NCX-1000 (2(acetyloxy) benzoic acid-3(nitrooxymethyl)phenyl ester) is an nitric oxide (NO)-releasing derivative of ursodeoxycholic acid (UDCA), which showed selective vasodilatory effect on intrahepatic circulation in animal models of cirrhosis. This study was aimed at testing the efficacy and tolerability of this compound in patients with cirrhosis and portal hypertension.

Clinical Outcomes Following Treatment of Non-contained Intrabony Defects with Enamel Matrix Derivative (EMD) or Guided Tissue Regeneration (GTR). A 12-month Randomized Controlled Clinical trial

The purpose of this study is to compare the healing of deep, non-contained intrabony defects (i.e., with a ?80% 1-wall component and a residual 2- to 3-wall component in the most apical part) treated with either an enamel matrix derivative (EMD) or guided tissue regeneration (GTR) after 12 months.

Healing of two and three wall intrabony periodontal defects following treatment with an enamel matrix derivative combined with autogenous bone

BACKGROUND: There are still limited data on the outcomes of regenerative periodontal surgery using a combination of an enamel matrix protein derivative (EMD) and autogenous bone (AB). AIM: To evaluate the healing of deep intrabony defects treated with either a combination EMD+AB or EMD alone. MATERIALS AND METHODS: Forty patients with advanced chronic periodontitis, with one deep intrabony defect, were randomly treated with either EMD+AB (test) or EMD (control). Clinical assessments were performed at baseline and at 1 year after treatment. The primary outcome variable was relative attachment level (RAL). RESULTS: Healing was uneventful in all patients. The test sites showed a reduction in the mean probing pocket depth (PPD) of 5.6 +/- 0.9 mm (p<0.001), a gain in the mean RAL of 4.2 +/- 1.1 mm (p<0.001) and a gain in the mean probing bone level (PBL) of 3.9 +/- 1.0 mm (p<0.001). The control group displayed a mean PPD reduction of 4.6 +/- 0.4 mm (p<0.001), a mean RAL gain of 3.4 +/- 0.8 mm (p<0.001) and a mean PBL gain of 2.8 +/- 0.8 mm (p<0.001). RAL gains of > or =4 mm were measured in 90% of the test defects and in 55% of the controls. PBL gains of > or =4 mm were obtained in 85% of the test defects and in 25% of the control ones. The test treatment resulted in statistically higher PPD reductions, RAL gains and PBL gains compared with the control (p<0.01). CONCLUSIONS: Within their limits, the present results indicate that: (i) at 1 year after surgery, both therapies resulted in statistically significant clinical improvements compared with baseline and (ii) although the combination of EMD+AB resulted in statistically significant higher soft and hard tissue improvements compared with treatment with EMD, the clinical relevance of this finding is unclear.

The quassinoid derivative NBT-272 targets both the AKT and ERK signaling pathways in embryonal tumors

The quassinoid analogue NBT-272 has been reported to inhibit MYC, thus warranting a further effort 7to better understand its preclinical properties in models of embryonal tumors (ET), a family of childhood malignancies sharing relevant biological and genetic features such as deregulated expression of MYC oncogenes. In our study, NBT-272 displayed a strong antiproliferative activity in vitro that resulted from the combination of diverse biological effects, ranging from G(1)/S arrest of the cell cycle to apoptosis and autophagy. The compound prevented the full activation of both eukaryotic translation initiation factor 4E (eIF4E) and its binding protein 4EBP-1, regulating cap-dependent protein translation. Interestingly, all responses induced by NBT-272 in ET could be attributed to interference with 2 main proproliferative signaling pathways, that is, the AKT and the MEK/extracellular signal-regulated kinase pathways. These findings also suggested that the depleting effect of NBT-272 on MYC protein expression occurred via indirect mechanisms, rather than selective inhibition. Finally, the ability of NBT-272 to arrest tumor growth in a xenograft model of neuroblastoma plays a role in the strong antitumor activity of this compound, both in vitro and in vivo, with its potential to target cell-survival pathways that are relevant for the development and progression of ET.

Effect of enamel matrix derivative and parathyroid hormone on bone formation in standardized osseous defects: an experimental study in minipigs

Previous experimental studies have indicated that locally administered enamel matrix derivative (EMD) and parathyroid hormone (PTH) may have a stimulatory effect on bone formation. However, it is not clear if the positive effect of EMD is related to its effect on the periodontium as a whole or directly on the bone-forming cells. In addition, it is not known if the presentation of PTH by adding the amino acid sequence Arg-Gly-Asp (RGD) is essential for its osteopromotive effect. Local delivery of a bioactive substance at the right time and in the right concentration often constitutes a major challenge. Polyethylene glycol-based hydrogel (PEG) is a degradable vehicle developed for delivery of bioactive proteins. To enhance the mechanical stability of the PEG-bioactive substance complex, an osteoconductive bone substitute material is often needed.