Introduction v Chapter 1. The Geometric setting 1.1. Cut-locus and volume growth function 1.2. Model manifolds and basic comparisons 1.3. Some spectral theory on manifolds Chapter 2. Some geometric examples related to oscillation theory 2.1. Conjugate points and Myers type compactness results 2.2. The spectrum of the Laplacian on complete manifolds 2.3. Spectral estimates and immersions 2.4. Spectral estimates and nonlinear PDE Chapter 3. On the solutions of the ODE (vz) + Avz = 0 3.1. Existence, uniqueness and the behaviour of zeroes 3.2. The critical curve: definition and main estimates Chapter 4. Below the critical curve 4.1. Positivity and estimates from below 4.2. Stability, index of −∆ − q(x) and the uncertainty principle 4.3. A comparison at infinity for nonlinear PDE 4.4. Yamabe type equations with a sign-changing nonlinearity 4.5. Upper bounds for the number of zeroes of z Chapter 5. Exceeding the critical curve 5.1. First zero and oscillation 5.2. Comparison with known criteria 5.3. Instability and index of −∆ − q(x) 5.4. Some remarks on minimal surfaces 5.5. Newton operators, unstable hypersurfaces and the Gauss map 5.6. Dealing with a possibly negative potential 5.7. An extension of Calabi compactness criterion Chapter 6. Much above the critical curve 6.1. Controlling the oscillation 6.2. The growth of the index of −∆ − q(x) 6.3. The essential spectrum of −∆ and punctured manifolds Bibliography

In this paper, we investigate the prescribed scalar curvature problem on a non-compact Riemannian manifold (M, , ), namely the existence of a conformal deformation of the metric , realizing a given function s(x) as its scalar curvature. In particular, the work focuses on the case when s(x) changes sign. Our main achievement are two new existence results requiring minimal assumptions on the underlying manifold, and ensuring a control on the stretching factor of the conformal deformation in such a way that the conformally deformed metric be bi-Lipschitz equivalent to the original one. The topological-geometrical requirements we need are all encoded in the spectral properties of the standard and conformal Laplacians of M . Our techniques can be extended to investigate the existence of entire positive solutions of quasilinear equations of the typewhere ∆p is the p-Laplacian, σ > p − 1 > 0, a, b ∈ L ∞ loc (M ) and b changes sign, and in the process of collecting the material for the proof of our theorems, we have the opportunity to give some new insight on the subcriticality theory for the Schrödinger type operatorIn particular, we prove sharp Hardy-type inequalities in some geometrically relevant cases, notably for minimal submanifolds of the hyperbolic space.

Defibrotide is a polynucleotide extracted from mammalian lung, which shows antithrombotic and anti-ischaemic activity in animals, probably related to stimulation of fibrinolysis and/or enhancement of prostacyclin production. The effect of a single infusion of defibrotide on fibrinolysis and the levels of certain prostanoids in man has been investigated in a cross-over double-blind placebo-controlled study. Evaluation of changes in fibrinolysis was difficult because of the spontaneous activation observed after placebo. However, the fast-acting plasminogen activator inhibitor was decreased only at end of the defibrotide infusion, suggesting a moderate profibrinolytic effect superimposed on the spontaneous activation. There was a marked and prolonged elevation of the plasma level of 6-keto-PGF1 alpha, the stable metabolite of prostacyclin. In collagen stimulated whole blood, both 6-keto-PGF1 alpha and prostaglandin E2 production were also greatly increased, with no consistent indication of inhibition of thromboxane B2. It is suggested that defibrotide stimulates prostacyclin and prostaglandin E2 production by leucocytes or via platelet/leukocyte interactions. The effects observed here should be useful in guiding subsequent clinical trials.

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We evaluated the effect of acupuncture on NSAID resistant dysmenorrhea related pain [measured according to Visual Analogue Scale (VAS)] in 15 consecutive patients. Pain was measured at baseline (T1), mid treatment (T2), end of treatment (T3) and 3 (T4) and 6 months (T5) after the end of treatment. Substantial reduction of pain and NSAID assumption was observed in 13 of 15 patients (87%). Pain intensity was significantly reduced with respect to baseline (average VAS = 8.5), by 64, 72, 60 or 53% at T2, T3, T4 or T5. Greater reduction of pain was observed for primary as compared with secondary dysmenorrhea. Average pain duration at baseline (2.6 days) was significantly reduced by 62, 69, 54 or 54% at T2, T3, T4 or T5. Average NSAID use was significantly reduced by 63, 74, 58 or 58% at T2, T3, T4 or T5, respectively, and ceased totally in 7 patients, still asymptomatic 6 months after treatment. Our findings suggest that acupuncture may be indicated to treat dysmenorrhea related pain, in particular in those subjects in whom NSAID or oral contraceptives are contraindicated or refused.