Dual quadratic differentials and entire minimal graphs in Heisenberg space

We define holomorphic quadratic differentials for spacelike surfaces with constant mean curvature in the Lorentzian homogeneous spaces $\mathbb{L}(\kappa,\tau)$ with isometry group of dimension 4, which are dual to the Abresch-Rosenberg differentials in the Riemannian counterparts $\mathbb{E}(\kappa,\tau)$, and obtain some consequences. On the one hand, we give a very short proof of the Bernstein problem in Heisenberg space, and provide a geometric description of the family of entire graphs sharing the same differential in terms of a 2-parameter conformal deformation. On the other hand, we prove that entire minimal graphs in Heisenberg space have negative Gauss curvature.Comment: 19 page

Random many-particle systems: applications from biology, and propagation of chaos in abstract models

The paper discusses a family of Markov processes that represent many particle systems, and their limiting behaviour when the number of particles go to infinity. The first part concerns model of biological systems: a model for sympatric speciation, i.e. the process in which a genetically homogeneous population is split in two or more different species sharing the same habitat, and models for swarming animals. The second part of the paper deals with abstract many particle systems, and methods for rigorously deriving mean field models.Comment: These are notes from a series of lectures given at the 5$^{th}$ Summer School on Methods and Models of Kinetic Theory, Porto Ercole, 2010. They are submitted for publication in "Rivista di Matematica della Universit\`a di Parma

Ordinal proportional cost sharing

We consider cost sharing problems with variable demands of heterogeneous goods. We study the compatibility of two axioms imposed on cost sharing methods: ordinality and average cost pricing for homogeneous (ACPH) goods. We generalize the ordinal proportional method (OPM) for the two-agent case, Sprumont [Journal of Economic Theory 81 (1998) 126162] to arbitrary number of agents

Second main theorem and unicity of meromorphic mappings for hypersurfaces of projective varieties in subgeneral position

The purpose of this article is twofold. The first is to prove a second main theorem for meromorphic mappings of \C^m into a complex projective variety intersecting hypersurfaces in subgeneral position with truncated counting functions. The second is to show a uniqueness theorem for these mappings which share few hypersurfaces without counting multiplicity.Comment: 16 page

Family networks and household outcomes in an economic crisis

Thesis (Ph.D.)--Boston UniversityThis thesis theoretically and empirically analyzes the nature and consequences of interactions between family members. The first chapter tests whether children's human capital accumulation was significantly affected by earnings shocks to their nonresident kin in the context of the 1997-8 financial crisis in Indonesia. The crisis produced sudden, heterogeneous shocks that facilitate the construction of an exogenous measure of earnings changes. Results indicate that earnings shocks to nonresident kin - including extended family and relatives living in other districts- significantly affected children's human capital accumulation between 1997 and 2000, and ultimate educational attainment measured nearly a decade after the crisis hit. Supplementary results point to intra-family transfers, underpinned by ex post altruism, as an important channel of causation. The second chapter develops a theoretical model of private transfers underpinned by ex post altruism among members of a network. I use this model to analyze equilibrium transfer patterns and inequality under alternative income distributions and network structures. I demonstrate the general intuition that transfers obtain in equilibrium when the amount of altruism is sufficiently strong relative to income inequality. Within the networks that I analyze, every equilibrium involving transfers takes the same form: unique income thresholds separate senders from receivers. Effective risk sharing takes place among senders and receivers, while those at intermediate incomes remain in autarky. Every equilibrium gives rise to the same set of allocations. I contrast these predictions with insurance-based theories of transfers in which risk sharing is operative for small in come differences and may fall apart at large income differences. The third chapter uses longitudinal data spanning nearly fifteen years to test whether transfers among family members within Indonesia are consistent with ex post altruism, against the alternative of insurance. I use the predicted effects of permanent versus transitory income on transfers, as well as theoretical predictions from the second chapter regarding the shape of transfer functions , to carry out this test. The results provide some evidence that transfer motives are inconsistent with insurance but consistent with ex post altruism

When does aggregation reduce uncertainty aversion?

We study the problem of uncertainty sharing within a household: "risk sharing," in a context of Knightian uncertainty. A household shares uncertain prospects using a social welfare function. We characterize the social welfare functions such that the household is collectively less averse to uncertainty than each member, and satises the Pareto principle and an independence axiom. We single out the sum of certainty equivalents as the unique member of this family which provides quasiconcave rankings over risk-free allocations