A note on the divergence-free Jacobian Conjecture in R^2

We give a shorter proof to a recent result by Neuberger, in the real case. Our result is essentially an application of the global asymptotic stability Jacobian Conjecture. We also extend some of the results presented in Neuberger

The period functions' higher order derivatives

We prove a formula for the $n$-th derivative of the period function $T$ in a period annulus of a planar differential system. For $n = 1$, we obtain Freire, Gasull and Guillamon formula for the period's first derivative \cite{FGG}. We apply such a result to hamiltonian systems with separable variables and other systems. We give some sufficient conditions for the period function of conservative second order O.D.E.'s to be convex.Comment: 2 figure

On the Chern numbers and the Hilbert polynomial of an almost complex manifold with a circle action

Let $(M,J)$ be a compact, connected, almost complex manifold of dimension $2n$ endowed with a $J$-preserving circle action with isolated fixed points. In this note we analyse the `geography problem' for such manifolds, deriving equations relating the Chern numbers to the index $k_0$ of $(M,J)$. We study the symmetries and zeros of the Hilbert polynomial associated to $(M,J)$, which imply many rigidity results for the Chern numbers when $k_0\neq 1$. We apply these results to the category of compact, connected symplectic manifolds. A long-standing question posed by McDuff and Salamon, also known as the `McDuff conjecture', asked about the existence of non-Hamiltonian actions with isolated fixed points. This question was answered recently by Tolman, with an explicit construction of a six-dimensional manifold with such an action. One issue that this raises is whether one can find topological criteria that ensure the manifold can only support a Hamiltonian or only a non-Hamiltonian action. In this vein, we are able to deduce such criteria from our rigidity theorems in terms of relatively few Chern numbers, depending on the index. This improves upon results of Feldman for which one needs to know the entire Todd genus. Another consequence of our work is that, in the situation above with a Hamiltonian action, the minimal Chern number coincides with the index and is at most $n+1$, mirroring results of Michelsohn in the complex category and Hattori in the almost complex category.Comment: v2. 38 pages, 1 figure. Major revision, especially in Section 6 (in particular, Proposition 6.8 contained an incorrect statement

The Role of Social Capital in Economic Development

This paper carries out an empirical assessment of the causal nexus connecting social capital's diverse aspects to the "quality" of economic development in Italy. The analysis accounts for three main social capital dimensions (i.e. bonding, bridging and linking social capital) and measures them through synthetic indicators built by means of principal component analyses performed on a dataset including multiple variables. The quality of development is measured through human development and indicators of the state of health of urban ecosystems, public services, social protection, gender equality, and labour markets. The causal relationship between social capital's and development's different dimensions is then assessed through structural equations models. The analysis in this paper provides a proof of Putnam's claims on the positive role of civil society organizations in development processes.Social capital; Social networks; Civil Society; Economic development; Social quality; Labour precariousness; Structural Equations Modelling

Work status and family planning: insights from the Italian puzzle

This paper uses a dataset built by the author on the basis of raw data taken from different national surveys to carry out an investigation into the socio-economic determinants of couples’ childbearing decisions in Italy. Since having children is in most cases a “couple matter”, the analysis accounts for the characteristics of both the aspiring parents. Our results contradict theoretical predictions according to which the increase in the opportunity cost of motherhood connected to higher female labour participation is responsible for the fall in fertility. On the contrary, the instability of the women’s work status (i.e. their being occasional, precarious, and low-paid workers) reveals to be a significant and strong dissuasive deterrent discouraging the decision to have children. Couples with unemployed women are less likely to plan childbearing as well. Other relevant explanatory variables are age, current family size, and the strength of family ties.Fertility; Family planning; Childbearing; Labour market; Female participation; Labour precariousness; Social capital; Italy