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.

Using a neural network approach for muon reconstruction and triggering

The extremely high rate of events that will be produced in the future Large Hadron Collider requires the triggering mechanism to take precise decisions in a few nano-seconds. We present a study which used an artificial neural network triggering algorithm and compared it to the performance of a dedicated electronic muon triggering system. Relatively simple architecture was used to solve a complicated inverse problem. A comparison with a realistic example of the ATLAS first level trigger simulation was in favour of the neural network. A similar architecture trained after the simulation of the electronics first trigger stage showed a further background rejection.Comment: A talk given at ACAT03, KEK, Japan, November 2003. Submitted to Nuclear Instruments and Methods in Physics Research, Section

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

Discrete approximations for complex Kac-Moody groups

We construct a map from the classifying space of a discrete Kac-Moody group over the algebraic closure of the field with p elements to the classifying space of a complex topological Kac-Moody group and prove that it is a homology equivalence at primes q different from p. This generalises a classical result of Quillen-Friedlander-Mislin for Lie groups. As an application, we construct unstable Adams operations for general Kac-Moody groups compatible with the Frobenius homomorphism. In contrast to the Lie case, the homotopy fixed points of these unstable Adams operations cannot be realized at q as the classifying spaces of Kac-Moody groups over finite fields. Our results rely on new integral homology decompositions for certain infinite dimensional unipotent subgroups of discrete Kac-Moody groups.Comment: New title and revised introduction, references added; results and proofs unchanged, 31 pages, 1 figur