L2L^2 estimates for the ∂ˉ\bar \partial operator

This is a survey article about $L^2$ estimates for the $\bar \partial$ operator. After a review of the basic approach that has come to be called the "Bochner-Kodaira Technique", the focus is on twisted techniques and their applications to estimates for $\bar \partial$, to $L^2$ extension theorems, and to other problems in complex analysis and geometry, including invariant metric estimates and the $\bar \partial$-Neumann Problem.Comment: To appear in Bulletin of Mathematical Science

Analytic inversion of adjunction: L^2 extension theorems with gain

We establish new results on weighted $L^2$ extension of holomorphic top forms with values in a holomorphic line bundle, from a smooth hypersurface cut out by a holomorphic function. The weights we use are determined by certain functions that we call denominators. We give a collection of examples of these denominators related to the divisor defined by the submanifold.Comment: To Appear in Ann. Inst. Fourie

Coordination of Purchasing and Bidding Activities Across Markets

In both consumer purchasing and industrial procurement, combinatorial interdependencies among the items to be purchased are commonplace. E-commerce compounds the problem by providing more opportunities for switching suppliers at low costs, but also potentially eases the problem by enabling automated market decision-making systems, commonly referred to as trading agents, to make purchasing decisions in an integrated manner across markets. Most of the existing research related to trading agents assumes that there exists a combinatorial market mechanism in which buyers (or sellers) can bid (or sell) service or merchant bundles. Todayâ??s prevailing e-commerce practice, however, does not support this assumption in general and thus limits the practical applicability of these approaches. We are investigating a new approach to deal with the combinatorial interdependency challenges for online markets. This approach relies on existing commercial online market institutions such as posted-price markets and various online auctions that sell single items. It uses trading agents to coordinate a buyerâ??s purchasing and bidding activities across multiple online markets simultaneously to achieve the best overall procurement effectiveness. This paper presents two sets of models related to this approach. The first set of models formalizes optimal purchasing decisions across posted-price markets with fixed transaction costs. Flat shipping costs, a common e-tailing practice, are captured in these models. We observe that making optimal purchasing decisions in this context is NP-hard in the strong sense and suggest several efficient computational methods based on discrete location theory. The second set of models is concerned with the coordination of bidding activities across multiple online auctions. We study the underlying coordination problem for a collection of first or second-price sealed-bid auctions and derive the optimal coordination and bidding policies.

Twenty Years of Timing SS433

We present observations of the optical ``moving lines'' in spectra of the Galactic relativistic jet source SS433 spread over a twenty year baseline from 1979 to 1999. The red/blue-shifts of the lines reveal the apparent precession of the jet axis in SS433, and we present a new determination of the precession parameters based on these data. We investigate the amplitude and nature of time- and phase-dependent deviations from the kinematic model for the jet precession, including an upper limit on any precessional period derivative of $\dot P < 5 \times 10^{-5}$. We also dicuss the implications of these results for the origins of the relativistic jets in SS433.Comment: 21 pages, including 9 figures. To appear in the Astrophysical Journa

Solitons supported by localized nonlinearities in periodic media

Nonlinear periodic systems, such as photonic crystals and Bose-Einstein condensates (BECs) loaded into optical lattices, are often described by the nonlinear Schr\"odinger/Gross-Pitaevskii equation with a sinusoidal potential. Here, we consider a model based on such a periodic potential, with the nonlinearity (attractive or repulsive) concentrated either at a single point or at a symmetric set of two points, which are represented, respectively, by a single {\delta}-function or a combination of two {\delta}-functions. This model gives rise to ordinary solitons or gap solitons (GSs), which reside, respectively, in the semi-infinite or finite gaps of the system's linear spectrum, being pinned to the {\delta}-functions. Physical realizations of these systems are possible in optics and BEC, using diverse variants of the nonlinearity management. First, we demonstrate that the single {\delta}-function multiplying the nonlinear term supports families of stable regular solitons in the self-attractive case, while a family of solitons supported by the attractive {\delta}-function in the absence of the periodic potential is completely unstable. We also show that the {\delta}-function can support stable GSs in the first finite gap in both the self-attractive and repulsive models. The stability analysis for the GSs in the second finite gap is reported too, for both signs of the nonlinearity. Alongside the numerical analysis, analytical approximations are developed for the solitons in the semi-infinite and first two finite gaps, with the single {\delta}-function positioned at a minimum or maximum of the periodic potential. In the model with the symmetric set of two {\delta}-functions, we study the effect of the spontaneous symmetry breaking of the pinned solitons. Two configurations are considered, with the {\delta}-functions set symmetrically with respect to the minimum or maximum of the potential

A review of High Performance Computing foundations for scientists

The increase of existing computational capabilities has made simulation emerge as a third discipline of Science, lying midway between experimental and purely theoretical branches [1, 2]. Simulation enables the evaluation of quantities which otherwise would not be accessible, helps to improve experiments and provides new insights on systems which are analysed [3-6]. Knowing the fundamentals of computation can be very useful for scientists, for it can help them to improve the performance of their theoretical models and simulations. This review includes some technical essentials that can be useful to this end, and it is devised as a complement for researchers whose education is focused on scientific issues and not on technological respects. In this document we attempt to discuss the fundamentals of High Performance Computing (HPC) [7] in a way which is easy to understand without much previous background. We sketch the way standard computers and supercomputers work, as well as discuss distributed computing and discuss essential aspects to take into account when running scientific calculations in computers.Comment: 33 page