Conformally invariant powers of the Laplacian, Q-curvature, and tractor calculus

We describe an elementary algorithm for expressing, as explicit formulae in tractor calculus, the conformally invariant GJMS operators due to C.R. Graham et alia. These differential operators have leading part a power of the Laplacian. Conformal tractor calculus is the natural induced bundle calculus associated to the conformal Cartan connection. Applications discussed include standard formulae for these operators in terms of the Levi-Civita connection and its curvature and a direct definition and formula for T. Branson's so-called Q-curvature (which integrates to a global conformal invariant) as well as generalisations of the operators and the Q-curvature. Among examples, the operators of order 4, 6, and 8 and the related Q-curvatures are treated explicitly. The algorithm exploits the ambient metric construction of Fefferman and Graham and includes a procedure for converting the ambient curvature and its covariant derivatives into tractor calculus expressions. This is partly based on "Standard tractors and the conformal ambient metric construction" (A. Cap and A.R. Gover, math.DG/0207016), where the relationship of the normal standard tractor bundle to the ambient construction is described.Comment: 42 pages. No figures. Record of changes: V1, 15 January 2002: Original posting. V2, 17 January 2002: Changing comment fields. Leaving abstract and text of article unchanged. V3, 1 February 2003: Minor changes and typographical corrections throughout articl

Tapering Enhanced Stimulated Superradiant Oscillator

In this paper, we present a new kind of high power and high efficiency free-electron laser oscillator based on the application of the tapering enhanced stimulated superradiant amplification (TESSA) scheme. The main characteristic of the TESSA scheme is a high intensity seed pulse which provides high gradient beam deceleration and efficient energy extraction. In the oscillator configuration, the TESSA undulator is driven by a high repetition rate electron beam and embedded in an optical cavity. A beam-splitter is used for outcoupling a fraction of the amplified power and recirculate the remainder as the intense seed for the next electron beam pulse. The mirrors in the oscillator cavity refocus the seed at the undulator entrance and monochromatize the radiation. In this paper we discuss the optimization of the system for a technologically relevant example at 1 $\mu$m using a 1~MHz repetition rate electron linac starting with an externally injected igniter pulse.Comment: 24 pages, 13 figure

Metric connections in projective differential geometry

We search for Riemannian metrics whose Levi-Civita connection belongs to a given projective class. Following Sinjukov and Mikes, we show that such metrics correspond precisely to suitably positive solutions of a certain projectively invariant finite-type linear system of partial differential equations. Prolonging this system, we may reformulate these equations as defining covariant constant sections of a certain vector bundle with connection. This vector bundle and its connection are derived from the Cartan connection of the underlying projective structure.Comment: 10 page

The Anti-Self-Dual Deformation Complex and a conjecture of Singer

Let $(M^4,g)$ be a smooth, closed, oriented anti-self-dual (ASD) four-manifold. $(M^4,g)$ is said to be unobstructed if the cokernel of the linearization of the self-dual Weyl tensor is trivial. This condition can also be characterized as the vanishing of the second cohomology group of the ASD deformation complex, and is central to understanding the local structure of the moduli space of ASD conformal structures. It also arises in construction of ASD manifolds by twistor and gluing methods. In this article we give conformally invariant conditions which imply an ASD manifold of positive Yamabe type is unobstructed

Einstein metrics in projective geometry

It is well known that pseudo-Riemannian metrics in the projective class of a given torsion free affine connection can be obtained from (and are equivalent to) the solutions of a certain overdetermined projectively invariant differential equation. This equation is a special case of a so-called first BGG equation. The general theory of such equations singles out a subclass of so-called normal solutions. We prove that non-degerate normal solutions are equivalent to pseudo-Riemannian Einstein metrics in the projective class and observe that this connects to natural projective extensions of the Einstein condition.Comment: 10 pages. Adapted to published version. In addition corrected a minor sign erro

Radiation measurements in the new tandem accelerator FEL

The measurements of both spontaneous and stimulated emissions of radiation in the newly configured Israeli EA-FEL are made for the first time. The radiation at the W-band was measured and characterized. The results match the predictions of our earlier theoretical modeling and calculations.Comment: 4 pages, 3 figures, FEL 2003 Conference repor

Rational Solutions for Challenges of the New Mellennium

We have reviewed ten major public problems challenging our Nation as it enters the new millennium. These are defense, healthcare costs, education, aging population, energy and environment, crime, low productivity growth services, income distribution, regulations, and infrastructure. These problems share several features. First, each is so large, if it were soIved; it would have major impact on the U.S. economy. Second, each is resident in a socioeconomic system containing non-linear feedback loops and an adaptive human element. Third, each can only be solved by our political system, yet these problems are not responsive to piecemeal problem solving, the approach traditionally used by policy makers. However, unless each problem is addressed in the context of the system in which it resides, the solution maybe worse than the problem. Our political system is immersed in reams of disconnected, unintelligible information skewed by various special interests to suggest policies favoring their particular needs. Help is needed, if rational solutions that serve public interests are to be forged for these ten probIems, The simulation and modeIing tools of physical scientists, engineers, economists, social scientists, public policy experts, and others, bolstered by the recent explosive growth in massively parallel computing power, must be blended together to synthesize models of the complex systems in which these problems are resident. These models must simulate the seemingly chaotic human element inherent in these systems and support policymakers in making informed decKlons about the future. We propose altering the policy development process by incorporating more modeling, simulation and analysis to bring about a revolution in policy making that takes advantage of the revolution in engineering emerging from simulation and modeling. While we recommend major research efforts to address each of these problems, we also observe these to be very complex, highly interdependent, multi-disciplinary problems; it will challenge the U.S. community of individual investigator researchers to make the cultural transformation necessary to address these problems in a team environment. Furthermore, models that simulate future behavior of these complex systems will not be exacq therefore, researchers must be prepared to use the modeling and simulation tools they develop to propose experiments to Congress. We recommend that ten laboratories owned by the American public be selected in an interagency competition to each manage and host a $1 billion/yertr National effort, each focused on one of these ten problems. Much of the supporting research and subsystem modeling work will be conducted at U.S. universities and at private firms with relevant expertise. Success of the Manhattan Project at the middle of the 20th century provides evidence this leadership model works