The stability and fates of hierarchical two-planet systems

We study the dynamical stability and fates of hierarchical (in semi-major axis) two-planet systems with arbitrary eccentricities and mutual inclinations. We run a large number of long-term numerical integrations and use the Support Vector Machine algorithm to search for an empirical boundary that best separates stable systems from systems experiencing either ejections or collisions with the star. We propose the following new criterion for dynamical stability: $a_{\rm out}(1-e_{\rm out})/[a_{\rm in}(1+e_{\rm in})]>2.4\left[\max(\mu_{\rm in},\mu_{\rm out})\right]^{1/3}(a_{\rm out}/a_{\rm in})^{1/2}+1.15$, which should be applicable to planet-star mass ratios $\mu_{\rm in},\mu_{\rm out}=10^{-4}-10^{-2}$, integration times up to $10^8$ orbits of the inner planet, and mutual inclinations $\lesssim40^\circ$. Systems that do not satisfy this condition by a margin of $\gtrsim0.5$ are expected to be unstable, mostly leading to planet ejections if $\mu_{\rm in}>\mu_{\rm out}$, while slightly favoring collisions with the star for $\mu_{\rm in}<\mu_{\rm out}$. We use our numerical integrations to test other stability criteria that have been proposed in the literature and show that our stability criterion performs significantly better for the range of system parameters that we have explored.Comment: 15 pages, 9 figures, to be published in the Astrophysical Journa

Self-propelled non-linearly diffusing particles. Aggregation and continuum description

We introduce a model of self-propelled particles carrying out a Brownian motion with a diffusion coefficient which depends on the local density of particles within a certain finite radius. Numerical simulations show that in a range of parameters the long-time spatial distribution of particles is non-homogeneous, and clusters can be observed. A number density equation, which explains the emergence of the aggregates and indicates the values of the parameters for which they appear, is derived. Numerical results of this continuum density equation are also shown.Comment: 5 pages, 5 figures. Major modifications. A new figure and some references added. Final version accepted for publication in Phys. Rev.

International convergence and local divergence

This work presents a north-south endogenous-growth model that reproduces some recent EU stylized facts: convergence between countries, divergence between the same countries, more spatial concentration of economic activity and higher growth rates. We claim that the ongoing technological reduction of transaction costs can conceivably spur those phenomena, specially if a regional productive duality within the less-developed countries were reinforced by a biased incidence of that fall in transaction costs. A key element is Grossman and Helpman's complementarity between innovation and imitation. The channels that allow for higher growth-rates are migrations and scale-effects in the industrialized regions of the poorest countries.

Trade and migration: a U-shaped transition in Eastern Europe

This paper proposes a 2-country 3-region economic geography model that can account for the most salient stylized facts experienced by Eastern European transition economies during the 1990s. In contrast to the existing literature, which has favored technological explanations, trade liberalization and factor mobility are the only driving forces. The model correctly predicts that in the first half of the decade trade liberalization led to divergence in GDP per capita, both between the West and the East and within the East. Consistent with the data, in the second half of the decade, internal labor mobility in the East reversed this process, and convergence became the dominant force. The model furthermore shows that the same U-shaped pattern applies to relative industrialization of West and East, although within the East the hinterland continued to lose industry throughout the decade.Trade liberalization; migration; convergence; welfare

Group orderings, dynamics, and rigidity

Let G be a countable group. We show there is a topological relationship between the space CO(G) of circular orders on G and the moduli space of actions of G on the circle; as well as an analogous relationship for spaces of left orders and actions on the line. In particular, we give a complete characterization of isolated left and circular orders in terms of strong rigidity of their induced actions of G on $S^1$ and R. As an application of our techniques, we give an explicit construction of infinitely many nonconjugate isolated points in the spaces CO(F_{2n}) of circular orders on free groups disproving a conjecture from Baik--Samperton, and infinitely many nonconjugate isolated points in the space of left orders on the pure braid group P_3, answering a question of Navas. We also give a detailed analysis of circular orders on free groups, characterizing isolated orders

Limits and dynamics of randomly connected neuronal networks

Networks of the brain are composed of a very large number of neurons connected through a random graph and interacting after random delays that both depend on the anatomical distance between cells. In order to comprehend the role of these random architectures on the dynamics of such networks, we analyze the mesoscopic and macroscopic limits of networks with random correlated connectivity weights and delays. We address both averaged and quenched limits, and show propagation of chaos and convergence to a complex integral McKean-Vlasov equations with distributed delays. We then instantiate a completely solvable model illustrating the role of such random architectures in the emerging macroscopic activity. We particularly focus on the role of connectivity levels in the emergence of periodic solutions

Skill-Upgrading and the Savings of Immigrants

This note derives positive and normative implications about the effects of immigration on welfare and the skill composition of the labor force in receiving economies. The main channel through which immigration affects labor-market outcomes is the availability of new loanable funds for investment, which results in endogenous skill-upgrading. Given their high training costs and their lifelong working period, immigrants self-select as net lenders, which facilitates the upgrading of both new generations of natives and migrants. Under sufficient altruism towards future generations, this induces a Pareto-improvement among the current generations of natives.