On the duality between p-Modulus and probability measures

Motivated by recent developments on calculus in metric measure spaces $(X,\mathsf d,\mathfrak m)$, we prove a general duality principle between Fuglede's notion of $p$-modulus for families of finite Borel measures in $(X,\mathsf d)$ and probability measures with barycenter in $L^q(X,\mathfrak m)$, with $q$ dual exponent of $p\in (1,\infty)$. We apply this general duality principle to study null sets for families of parametric and non-parametric curves in $X$. In the final part of the paper we provide a new proof, independent of optimal transportation, of the equivalence of notions of weak upper gradient based on $p$-Modulus (Koskela-MacManus '98, Shanmugalingam '00) and suitable probability measures in the space of curves (Ambrosio-Gigli-Savare '11)Comment: Minor corrections, typos fixe

Duality properties of metric Sobolev spaces and capacity

We study the properties of the dual Sobolev space H-1;q(X) = - H1;p(X) '0 on a complete extended metric-topological measure space X = (X; ⊤; d;m) for p 2 (1;1). We will show that a crucial role is played by the strong closure H-1;q pd (X) of Lq(X;m) in the dual H-1;q(X), which can be identified with the predual of H1;p(X). We will show that positive functionals in H-1;q(X) can be represented as a positive Radon measure and we will charaterize their dual norm in terms of a suitable energy functional on nonparametric dynamic plans. As a byproduct, we will show that for every Radon measure μ with finite dual Sobolev energy, Capp-negligible sets are also μ-negligible and good representatives of Sobolev functions belong to L1(X; μ). We eventually show that the Newtonian-Sobolev capacity Capp admits a natural dual representation in terms of such a class of Radon measures

Nonlinear diffusion equations and curvature conditions in metric measure spaces

The aim of this paper is to provide new characterizations of the curvature dimension condition in the context of metric measure spaces (X,d,m). On the geometric side, our new approach takes into account suitable weighted action functionals which provide the natural modulus of K-convexity when one investigates the convexity properties of N-dimensional entropies. On the side of diffusion semigroups and evolution variational inequalities, our new approach uses the nonlinear diffusion semigroup induced by the N-dimensional entropy, in place of the heat flow. Under suitable assumptions (most notably the quadraticity of Cheeger's energy relative to the metric measure structure) both approaches are shown to be equivalent to the strong CD∗(K,N) condition of Bacher-Sturm

Ricerca Sul Valore Economico Del Terzo Settore in Italia

This report summarizes the findings from a 2012 survey on the economic value of the third sector in Italy. Research includes snapshots of Italian non-profit and growth trends from recent years

Metric measure spaces with Riemannian Ricci curvature bounded from below

In this paper we introduce a synthetic notion of Riemannian Ricci bounds from below for metric measure spaces (X,d,m) which is stable under measured Gromov-Hausdorff convergence and rules out Finsler geometries. It can be given in terms of an enforcement of the Lott, Sturm and Villani geodesic convexity condition for the entropy coupled with the linearity of the heat flow. Besides stability, it enjoys the same tensorization, global-to-local and local-to-global properties. In these spaces, that we call RCD(K,\infty) spaces, we prove that the heat flow (which can be equivalently characterized either as the flow associated to the Dirichlet form, or as the Wasserstein gradient flow of the entropy) satisfies Wasserstein contraction estimates and several regularity properties, in particular Bakry-Emery estimates and the L^\infty-Lip Feller regularization. We also prove that the distance induced by the Dirichlet form coincides with d, that the local energy measure has density given by the square of Cheeger's relaxed slope and, as a consequence, that the underlying Brownian motion has continuous paths. All these results are obtained independently of Poincar\'e and doubling assumptions on the metric measure structure and therefore apply also to spaces which are not locally compact, as the infinite-dimensional onesComment: (v2) Minor typos, proof of Proposition 2.3, proof of Theorem 4.8: corrected. Proof of Theorem 6.2: corrected and simplified, thanks to the new Lemma 2.8. Lemma 3.6 and 4.6 (of v1) removed, since no more neede

Nonlinear diffusion equations and curvature conditions in metric measure spaces

The aim of this paper is to provide new characterizations of the curvature dimension condition in the context of metric measure spaces (X,d,m). On the geometric side, our new approach takes into account suitable weighted action functionals which provide the natural modulus of K-convexity when one investigates the convexity properties of N-dimensional entropies. On the side of diffusion semigroups and evolution variational inequalities, our new approach uses the nonlinear diffusion semigroup induced by the N-dimensional entropy, in place of the heat flow. Under suitable assumptions (most notably the quadraticity of Cheeger's energy relative to the metric measure structure) both approaches are shown to be equivalent to the strong CD∗(K,N) condition of Bacher-Sturm

Sudden Death in Indeterminate Chagas disease is Uncommon. A Systematic Review

La enfermedad de Chagas es un problema sanitario de gran magnitud en América Latina debido a su alta prevalencia, morbilidad y mortalidad. A su vez, las migraciones desde los países latinoamericanos hacia los Estados Unidos y Europa han dispersado a una cantidad significativa de personas portadoras de la enfermedad. Es importante tener en cuenta que la mayoría de los pacientes permanecen en la forma indeterminada de la enfermedad por décadas, sin manifestar ningún síntoma ni signo de su afección. A pesar de ello, hay quienes sostienen que la forma indeterminada conlleva un aumento del riesgo de padecer muerte súbita, aunque no hay estudios que se hayan diseñado específicamente a fin de esclarecer esta cuestión. En una revisión sistemática de los estudios con seguimiento de pacientes chagásicos asintomáticos con ECG normal y de causa conocida de muerte encontramos 15 artículos que incluyen el seguimiento de 9.382 pacientes. La mortalidad entre los asintomáticos con ECG normal fue muy baja (0,92%), que no resultó estadísticamente diferente de la de los controles no chagásicos (p = 0,38). Esta revisión sistemática muestra que la muerte súbita es infrecuente en la forma indeterminada. Estos pacientes tienen el mismo riesgo que la población general y por lo tanto se les debe permitir que lleven una vida normal tanto en el aspecto laboral como en lo relativo a su actividad física, sin alarmarlos innecesariamente sobre su condición clínica ni abrumarlos con estudios sofisticados y costosos. Es necesario, sin embargo, el control periódico, ya que si el paciente progresa a la forma cardíaca el riesgo de muerte aumenta notablemente.Chagas disease is a serious health care problem in Latin America due to its high prevalence, morbidity and mortality. The migration from Latin American countries to the United States and Europe has disseminated a significant number of infected subjects. Most patients present the indeterminate form of the disease and remain without symptoms for decades. However, some groups believe that patients with the indeterminate form are at high risk for developing sudden death although no studies have been designed to investigate this issue. We conducted a systematic review of follow-up studies in patients with asymptomatic Chagas disease, normal ECG and known cause of death. We found 15 articles including 9382 patients. Mortality rate in asymptomatic patients with normal ECG was very low (0.92%) and similar to that of controls without Chagas disease (p=0.38). This systematic review shows that sudden death is uncommon in the indeterminate form of the disease and that the risk of death is similar to that of the general population. Thus, these patients should be allowed to lead a normal working life and to practice physical activity, without alarming them unnecessarily about their condition or indicating sophisticated and expensive studies. Regular follow-up is necessary as the death risk increases considerably when the disease progresses to the cardiac form.Fil: Gonzalez, Julian. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Houssay. Instituto de Investigaciones Cardiológicas (i); ArgentinaFil: Azzato, Francisco. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Houssay. Instituto de Investigaciones Cardiológicas (i); ArgentinaFil: Ambrosio, Giuseppe. Universita Di Perugia; ItaliaFil: Milei, Jose. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Houssay. Instituto de Investigaciones Cardiológicas (i); Argentin