Infinitesimal Variations of Hodge Structure at Infinity

By analyzing the local and infinitesimal behavior of degenerating polarized variations of Hodge structure the notion of infinitesimal variation of Hodge structure at infinity is introduced. It is shown that all such structures can be integrated to polarized variations of Hodge structure and that, conversely, all are limits of infinitesimal variations of Hodge structure (IVHS) at finite points. As an illustration of the rich information encoded in this new structure, some instances of the maximal dimension problem for this type of infinitesimal variation are presented and contrasted with the "classical" case of IVHS at finite points

Trajectories of Couple Relationship Quality after Childbirth: Does Marriage Matter?

Marital quality typically declines following the birth of a (first) child, as parenthood brings new identities and responsibilities for mothers and fathers. Yet, it is unclear whether nonmarital relationship quality follows a similar trajectory. This paper uses data from the Fragile Families and Child Wellbeing Study (N=2,500) with latent growth curve and difference-in-difference models to examine relationship quality for co-resident couples over five years after a child’s birth. Findings suggest that marriage at the time of the birth is protective for couple relationship quality, net of various individual characteristics associated with marriage. Among cohabiting couples at birth, those who subsequently marry have better relationship quality compared to all who do not marry, though not compared to stably cohabiting couples.

Highly Optimized Tolerance: Robustness and Power Laws in Complex Systems

We introduce highly optimized tolerance (HOT), a mechanism that connects evolving structure and power laws in interconnected systems. HOT systems arise, e.g., in biology and engineering, where design and evolution create complex systems sharing common features, including (1) high efficiency, performance, and robustness to designed-for uncertainties, (2) hypersensitivity to design flaws and unanticipated perturbations, (3) nongeneric, specialized, structured configurations, and (4) power laws. We introduce HOT states in the context of percolation, and contrast properties of the high density HOT states with random configurations near the critical point. While both cases exhibit power laws, only HOT states display properties (1-3) associated with design and evolution.Comment: 4 pages, 2 figure

Highly Optimized Tolerance: Robustness and Design in Complex Systems

Highly optimized tolerance (HOT) is a mechanism that relates evolving structure to power laws in interconnected systems. HOT systems arise where design and evolution create complex systems sharing common features, including (1) high efficiency, performance, and robustness to designed-for uncertainties, (2) hypersensitivity to design flaws and unanticipated perturbations, (3) nongeneric, specialized, structured configurations, and (4) power laws. We study the impact of incorporating increasing levels of design and find that even small amounts of design lead to HOT states in percolation

Power Laws, Highly Optimized Tolerance, and Generalized Source Coding

We introduce a family of robust design problems for complex systems in uncertain environments which are based on tradeoffs between resource allocations and losses. Optimized solutions yield the “robust, yet fragile” features of highly optimized tolerance and exhibit power law tails in the distributions of events for all but the special case of Shannon coding for data compression. In addition to data compression, we construct specific solutions for world wide web traffic and forest fires, and obtain excellent agreement with measured data