A mathematical theory of stochastic microlensing. II. Random images, shear, and the Kac-Rice formula

Continuing our development of a mathematical theory of stochastic microlensing, we study the random shear and expected number of random lensed images of different types. In particular, we characterize the first three leading terms in the asymptotic expression of the joint probability density function (pdf) of the random shear tensor due to point masses in the limit of an infinite number of stars. Up to this order, the pdf depends on the magnitude of the shear tensor, the optical depth, and the mean number of stars through a combination of radial position and the star's mass. As a consequence, the pdf's of the shear components are seen to converge, in the limit of an infinite number of stars, to shifted Cauchy distributions, which shows that the shear components have heavy tails in that limit. The asymptotic pdf of the shear magnitude in the limit of an infinite number of stars is also presented. All the results on the random microlensing shear are given for a general point in the lens plane. Extending to the general random distributions (not necessarily uniform) of the lenses, we employ the Kac-Rice formula and Morse theory to deduce general formulas for the expected total number of images and the expected number of saddle images. We further generalize these results by considering random sources defined on a countable compact covering of the light source plane. This is done to introduce the notion of global expected number of positive parity images due to a general lensing map. Applying the result to microlensing, we calculate the asymptotic global expected number of minimum images in the limit of an infinite number of stars, where the stars are uniformly distributed. This global expectation is bounded, while the global expected number of images and the global expected number of saddle images diverge as the order of the number of stars. © 2009 American Institute of Physics.

Reexamining the science of marine protected areas: Linking knowledge to action

Marine protected areas (MPAs) are often implemented to conserve or restore species, fisheries, habitats, ecosystems, and ecological functions and services; buffer against the ecological effects of climate change; and alleviate poverty in coastal communities. Scientific research provides valuable insights into the social and ecological impacts of MPAs, as well as the factors that shape these impacts, providing useful guidance or "rules of thumb" for science-based MPA policy. Both ecological and social factors foster effective MPAs, including substantial coverage of representative habitats and oceanographic conditions; diverse size and spacing; protection of habitat bottlenecks; participatory decisionmaking arrangements; bounded and contextually appropriate resource use rights; active and accountable monitoring and enforcement systems; and accessible conflict resolution mechanisms. For MPAs to realize their full potential as a tool for ocean governance, further advances in policy-relevant MPA science are required. These research frontiers include MPA impacts on nontarget and wide-ranging species and habitats; impacts beyond MPA boundaries, on ecosystem services, and on resource-dependent human populations, as well as potential scale mismatches of ecosystem service flows. Explicitly treating MPAs as "policy experiments" and employing the tools of impact evaluation holds particular promise as a way for policy-relevant science to inform and advance science-based MPA policy. © 2011 Wiley Periodicals, Inc.

Tectonic and magmatic segmentation of the Global Ocean Ridge System: A synthesis of observations

© 2016 The Author(s).Mid-ocean ridges display tectonic segmentation defined by discontinuities of the axial zone, and geophysical and geochemical observations suggest segmentation of the underlying magmatic plumbing system. Here, observations of tectonic and magmatic segmentation at ridges spreading from fast to ultraslow rates are reviewed in light of influential concepts of ridge segmentation, including the notion of hierarchical segmentation, spreading cells and centralized v. multiple supply of mantle melts. The observations support the concept of quasi-regularly spaced principal magmatic segments, which are 30-50 km long on average at fast- to slow-spreading ridges and fed by melt accumulations in the shallow asthenosphere. Changes in ridge properties approaching or crossing transform faults are often comparable with those observed at smaller offsets, and even very small discontinuities can be major boundaries in ridge properties. Thus, hierarchical segmentation models that suggest large-scale transform fault-bounded segmentation arises from deeper level processes in the asthenosphere than the finer-scale segmentation are not generally supported. The boundaries between some but not all principal magmatic segments defined by ridge axis geophysical properties coincide with geochemical boundaries reflecting changes in source composition or melting processes. Where geochemical boundaries occur, they can coincide with discontinuities of a wide range of scales.

Design of Scheduling Algorithms Using Game Theoretic Ideas

Scheduling a set of jobs over a collection of machines to optimize a certain quality-of-service measure is one of the most important research topics in both computer science theory and practice. In this thesis, we design algorithms that optimize {\em flow-time} (or delay) of jobs for scheduling problems that arise in a wide range of applications. We consider the classical model of unrelated machine scheduling and resolve several long standing open problems; we introduce new models that capture the novel algorithmic challenges in scheduling jobs in data centers or large clusters; we study the effect of selfish behavior in distributed and decentralized environments; we design algorithms that strive to balance the energy consumption and performance.

The technically interesting aspect of our work is the surprising connections we establish between approximation and online algorithms, economics, game theory, and queuing theory. It is the interplay of ideas from these different areas that lies at the heart of most of the algorithms presented in this thesis.

The main contributions of the thesis can be placed in one of the following categories.

1. Classical Unrelated Machine Scheduling: We give the first polygorithmic approximation algorithms for minimizing the average flow-time and minimizing the maximum flow-time in the offline setting. In the online and non-clairvoyant setting, we design the first non-clairvoyant algorithm for minimizing the weighted flow-time in the resource augmentation model. Our work introduces iterated rounding technique for the offline flow-time optimization, and gives the first framework to analyze non-clairvoyant algorithms for unrelated machines.

2. Polytope Scheduling Problem: To capture the multidimensional nature of the scheduling problems that arise in practice, we introduce Polytope Scheduling Problem (\psp). The \psp problem generalizes almost all classical scheduling models, and also captures hitherto unstudied scheduling problems such as routing multi-commodity flows, routing multicast (video-on-demand) trees, and multi-dimensional resource allocation. We design several competitive algorithms for the \psp problem and its variants for the objectives of minimizing the flow-time and completion time. Our work establishes many interesting connections between scheduling and market equilibrium concepts, fairness and non-clairvoyant scheduling, and queuing theoretic notion of stability and resource augmentation analysis.

3. Energy Efficient Scheduling: We give the first non-clairvoyant algorithm for minimizing the total flow-time + energy in the online and resource augmentation model for the most general setting of unrelated machines.

4. Selfish Scheduling: We study the effect of selfish behavior in scheduling and routing problems. We define a fairness index for scheduling policies called {\em bounded stretch}, and show that for the objective of minimizing the average (weighted) completion time, policies with small stretch lead to equilibrium outcomes with small price of anarchy. Our work gives the first linear/ convex programming duality based framework to bound the price of anarchy for general equilibrium concepts such as coarse correlated equilibrium.