Strichartz estimates for the Schr\"odinger equation on polygonal domains

We prove Strichartz estimates with a loss of derivatives for the Schr\"odinger equation on polygonal domains with either Dirichlet or Neumann homogeneous boundary conditions. Using a standard doubling procedure, estimates the on polygon follow from those on Euclidean surfaces with conical singularities. We develop a Littlewood-Paley squarefunction estimate with respect to the spectrum of the Laplacian on these spaces. This allows us to reduce matters to proving estimates at each frequency scale. The problem can be localized in space provided the time intervals are sufficiently small. Strichartz estimates then follow from a result of the second author regarding the Schr\"odinger equation on the Euclidean cone.Comment: 12 page

Observation of enhanced optical spring damping in a macroscopic mechanical resonator and application for parametric instability control in advanced gravitational-wave detectors

We show that optical spring damping in an optomechanical resonator can be enhanced by injecting a phase delay in the laser frequency-locking servo to rotate the real and imaginary components of the optical spring constant. This enhances damping at the expense of optical rigidity. We demonstrate enhanced parametric damping which reduces the Q factor of a 0.1-kg-scale resonator from 1.3×10^5 to 6.5×10^3. By using this technique adequate optical spring damping can be obtained to damp parametric instability predicted for advanced laser interferometer gravitational-wave detectors

Rethinking Antitrust Injury

Substantive changes in antitrust law since 1977 have had a dramatic impact on the vitality of antitrust enforcement.\u27 Recent procedural changes now seem likely to have as great an influence. In the procedural area, the emphasis has been on antitrust standing and anti-trust injury. As a result of recent judicial interpretations of these requirements, antitrust plaintiffs face increasingly formidable hurdles. As courts focus on questions of standing and injury, important discussions about whether a practice should be held to a per se or rule of reason standards frequently are immaterial. If there is no qualified plaintiff,the substantive issue need never be addressed. Technically, standing requirements limit the array of potential plaintiffs while antitrust injury requirements limit the types of compensable harms. Together, however, they form a generalized standing requirement: a list of conditions a plaintiff must satisfy before qualifying to proceed to the substantive antitrust question. The Supreme Court neglected the issue until 1977. Since that time, the Court has considered the issue on five occasions. Despite this repeated analysis, the Court has provided little guidance with respect to antitrust standing.This lack of guidance can be traced to the Supreme Court\u27s enunciation of rather general guidelines and then applying those guidelines in substantive factual contexts that are not representative of most antitrust litigation. Consequently, a division between circuits has emerged with respect to the status of different classes of antitrust plaintiffs.\u2

Albrecht After ARCO: Maximum Resale Price Fixing Moves Toward the Rule of Reason

For some time, both economic and legal commentators have recognized the economic irrationality of the Supreme Court\u27s ruling in Albrecht v. Herald Co. which prohibited the imposition of maximum resale prices by a supplier on its resellers. Ordinarily, unwise decisions receive critical reviews and eventually lose their force as they are over-ruled explicitly or by implication in subsequent decisions. In order for this evolution to occur, however, the Court must be presented with an opportunity to alter its earlier rulings. Recently, the Supreme Court had just such an opportunity to revisit the Albrecht rule in Atlantic Richfield Co. v. USA Petroleum (ARCO). Although the Court\u27s reasoning in ARCO severely restricts some private suits, the Court refused to overturn Albrecht directly and left standing the shell of the per se doctrine.\u27 This is unfortunate because the Albrecht precedent remains on the books even though it is at odds with the promotion of consumer welfare. Two undesirable consequences follow. First, businesses avoid some procompetitive business arrangements for fear of antitrust liability. Given that the purpose of the antitrust laws is to promote competition, this is particularly perverse. Second, the lower courts are forced to rely on standing rules as a means of rejecting claims presented by undeserving plaintiffs. In this Article we assess the implications of ARCO for the future vitality of Albrecht. In Part II we review the development and economic consequences of the Albrecht rule. In Part III we examine the judicial hostility that has developed in the lower courts toward the anticompetitive nature of the Albrecht rule. In Part IV we analyze the ARCO decision with respect to substantive antitrust policy and antitrust injury