Lessons from \u3ci\u3eAmex\u3c/i\u3e for Platform Antitrust Litigation

I. Introduction II. A Brief History of a Long Litigation ... A. The District Court Rules for Plaintiffs Based on a One-Sided Market Definition ... B. The Second Circuit Reverses, Applying a Two-Sided Market Definition ... C. The Supreme Court Affirms, Embracing a Two-Sided Market Definition III. Lessons from Amex ... A. Lesson One: A Full Rule of Reason Analysis—Not Some Form of Relaxed Review Advocated by the Government—Applies to Vertical Agreements Between a Platform and Customers on One Side of the Platform ... B. Lesson Two: For That Rule of Reason Analysis, a Plaintiff First Must Define a Relevant Market That Includes Both Sides of Two-Sided Transaction Platforms C. Lesson Three: When the Relevant Market Is Two-Sided, a Plaintiff Must Demonstrate That the Challenged Conduct Harmed Competition in the Market as a Whole IV. Misperceptions About Amex ... A. Fallacy One: Two-Sided Market Definition Does Not Apply to “Mature” Platforms ... B. Fallacy Two: A Platform’s Conduct Should Be Condemned If Platform Consumers on One Side Are “Subsidized” by Those Who Do Not Use the Platform … C. Fallacy Three: Amex Will Complicate and Confuse Antitrust Analysis in a Wide Range of Case

Jet quenching in strongly coupled plasma

We present calculations in which an energetic light quark shoots through a finite slab of strongly coupled ${\cal N}=4$ supersymmetric Yang-Mills (SYM) plasma, with thickness $L$, focussing on what comes out on the other side. We find that even when the "jets" that emerge from the plasma have lost a substantial fraction of their energy they look in almost all respects like "jets" in vacuum with the same reduced energy. The one possible exception is that the opening angle of the "jet" is larger after passage through the slab of plasma than before. Along the way, we obtain a fully geometric characterization of energy loss in the strongly coupled plasma and show that $dE_{\rm out}/dL \propto L^2/\sqrt{x^2_{\rm stop}-L^2}$, where $E_{\rm out}$ is the energy of the "jet" that emerges from the slab of plasma and $x_{\rm stop}$ is the (previously known) stopping distance for the light quark in an infinite volume of plasma.Comment: 13 pages, 5 figure

On the Evolution of Jet Energy and Opening Angle in Strongly Coupled Plasma

We calculate how the energy and the opening angle of jets in ${\cal N}=4$ SYM theory evolve as they propagate through the strongly coupled plasma of that theory. We define the rate of energy loss $dE_{\rm jet}/dx$ and the jet opening angle in a straightforward fashion directly in the gauge theory before calculating both holographically, in the dual gravitational description. In this way, we rederive the previously known result for $dE_{\rm jet}/dx$ without the need to introduce a finite slab of plasma. We obtain a striking relationship between the initial opening angle of the jet, which is to say the opening angle that it would have had if it had found itself in vacuum instead of in plasma, and the thermalization distance of the jet. Via this relationship, we show that ${\cal N}=4$ SYM jets with any initial energy that have the same initial opening angle and the same trajectory through the plasma experience the same fractional energy loss. We also provide an expansion that describes how the opening angle of the ${\cal N}=4$ SYM jets increases slowly as they lose energy, over the fraction of their lifetime when their fractional energy loss is not yet large. We close by looking ahead toward potential qualitative lessons from our results for QCD jets produced in heavy collisions and propagating through quark-gluon plasma.Comment: 40 pages, 9 figures, v2: minor clarifications adde

The application of generalized, cyclic, and modified numerical integration algorithms to problems of satellite orbit computation

Generalized, cyclic, and modified multistep numerical integration methods are developed and evaluated for application to problems of satellite orbit computation. Generalized methods are compared with the presently utilized Cowell methods; new cyclic methods are developed for special second-order differential equations; and several modified methods are developed and applied to orbit computation problems. Special computer programs were written to generate coefficients for these methods, and subroutines were written which allow use of these methods with NASA's GEOSTAR computer program