High-momentum tail in the Tonks gas under harmonic confinement

We use boson-fermion mapping to show that the single-particle momentum distribution in a one-dimensional gas of hard point-like bosons (Tonks gas) inside a harmonic trap decays as $p^{-4}$ at large momentum $p$. The relevant integrals expressing the one-body density matrix are evaluated for small numbers of particles in a simple Monte Carlo approach to test the extent of the asymptotic law and to illustrate the slow decay of correlations between the matter-wave field at different points.Comment: 8 pages, 3 figures, accepted for publication in Phys. Lett.

A comparison of sample-based Stochastic Optimal Control methods

In this paper, we compare the performance of two scenario-based numerical methods to solve stochastic optimal control problems: scenario trees and particles. The problem consists in finding strategies to control a dynamical system perturbed by exogenous noises so as to minimize some expected cost along a discrete and finite time horizon. We introduce the Mean Squared Error (MSE) which is the expected $L^2$-distance between the strategy given by the algorithm and the optimal strategy, as a performance indicator for the two models. We study the behaviour of the MSE with respect to the number of scenarios used for discretization. The first model, widely studied in the Stochastic Programming community, consists in approximating the noise diffusion using a scenario tree representation. On a numerical example, we observe that the number of scenarios needed to obtain a given precision grows exponentially with the time horizon. In that sense, our conclusion on scenario trees is equivalent to the one in the work by Shapiro (2006) and has been widely noticed by practitioners. However, in the second part, we show using the same example that, by mixing Stochastic Programming and Dynamic Programming ideas, the particle method described by Carpentier et al (2009) copes with this numerical difficulty: the number of scenarios needed to obtain a given precision now does not depend on the time horizon. Unfortunately, we also observe that serious obstacles still arise from the system state space dimension

Disintegration

The silver lining behind the Supreme Court\u27s decision to disintegrate the Seattle and Louisville public schools is that the decision also runs the risk of disintegrating judicial review. Parents Involved in Community Schools v. Seattle School District No. 1 holds that the Constitution bars voluntary, race-conscious efforts by two local school boards to retain the racial integration that they worked so hard to achieve after Brown. In so holding, the Court curiously reads the Equal Protection Clause as preventing the use of race to pursue actual equality, and instead insists on a type of formal equality that has historically been associated with thinly veiled efforts to disguise racial oppression--the type of oppression that the Court authorized in upholding the separate-but-equal regime of Plessy. By using the Constitution to protect passive resegregation from active integration, the current Court ends up constitutionalizing the culture\u27s regression to the days of greater racial separation--a separation that Brown found to be inherently unequal. As a result, the new Resegregation decision has not only realigned the current Court with its own racially oppressive past, but it has also distanced the Court from the nation\u27s hope for a racially progressive future. Once the decision is understood in this way, the question becomes whether the case will begin to undermine the legitimacy needed for the Court to continue its activist conception of judicial review. Because the views of the Justices seem so transparently political, the threat to judicial legitimacy that emanates from the Resegregation case may end up exceeding the nation\u27s patience for continued Supreme Court interference in the nation\u27s racial policymaking process. There can be no assurance that the case will prompt such a reconsideration of judicial review. Part I of this article describes the manner in which the Resegregation decision has marginalized the importance of racial integration. Part I.A. describes the Seattle and Louisville integration plans under consideration in the case. Part I.B. describes the various Supreme Court opinions issued in the decision invalidating those plans. Part II discusses the impact that the Resegregation decision is likely to have on the nation\u27s ever-evolving conception of equality. Part II.A. explains how the decision effectively overrules Brown--by protecting the interests of disappointed white parents at the cost of advancing racial resegregation--despite the fact that it is doctrinally difficult to support such a result. Part lI.B. argues that the plurality opinion of Chief Justice Roberts now gives official recognition to an updated form of racism, in which supposed equality is used as a tool of racial oppression. Part III discusses the effect that the decision is likely to have on the future of judicial review. Part III.A. illustrates that the decision to invalidate the integration plans at issue can best be understood as political rather than doctrinal in nature. Part III.B. expresses the hope that such transparent judicial politics will cause the Supreme Court to lose the perceived legitimacy that it needs to continue supplanting the racial policy preferences adopted by the representative branches of government. The conclusion suggests that, while one may hope for the disintegration of undemocratically activist judicial review, the long persistence of racial oppression in the United States does not afford much basis for optimism in achieving that end

Interference of a thermal Tonks gas on a ring

A nonzero temperature generalization of the Fermi-Bose mapping theorem is used to study the exact quantum statistical dynamics of a one-dimensional gas of impenetrable bosons on a ring. We investigate the interference produced when an initially trapped gas localized on one side of the ring is released, split via an optical-dipole grating, and recombined on the other side of the ring. Nonzero temperature is shown not to be a limitation to obtaining high visibility fringes.Comment: 4 pages, 3 figure

Functional Analysis of the Plain-Error Rule

In this article, I attempt to do two things at once. First, I attempt to analyze the Supreme Court\u27s jurisdiction to conduct plain-error review of state court decisions. The plain-error issue merits consideration not only because of its intrinsic interest and arguable complexity, but also because the question whether the Supreme Court is authorized to engage in plain-error review is an open one that I would like to help resolve. My second objective, however, is the more important of the two. In the context of analyzing plain-error review, what I really want to do is analyze legal analysis itself. There are a variety of ways to approach legal problems, which can be usefully divided into two categories. One category strikes me as sensible, the other as silly. In the process of analyzing the plain-error issue, I hope to demonstrate which is which

Spinning the Legislative Veto

I am delighted to have been given the opportunity to comment on Judge Breyer\u27s proposal for a fast-track substitute to the legislative veto. Although the Supreme Court invalidated the legislative veto device in INS v. Chadha, Judge Breyer\u27s proposal demonstrates that innovative thinking may well permit those with enough determination to circumvent the apparent effect of the Court\u27s decision. Even more important, the proposal illustrates why such circumvention is possible. As a doctrinal matter, the legislative veto poses a real dilemma--one that is rooted in fundamental uncertainty about the proper relationship between the Supreme Court and the elected branches of government. Because the Supreme Court tried to resolve the constitutional issues raised by the legislative veto without first resolving that dilemma, the Chadha opinion is unsatisfying. Its tone is glib; its reasoning is superficial; and its analysis is linguistic rather than functional in nature

Neutralizing Grutter

Part I of this article argues that the Supreme Court lacks the institutional competence to formulate racial policy for the nation, and highlights the tension that exists between the Court\u27s abstract preference for race neutrality and the concrete reality of contemporary race relations, in which dedicated efforts to promote racial balance offer the only meaningful hope of eliminating systemic discrimination. Part II discusses moderate strategies that can be used to deflect the impact of Grutter’s prohibition on racial balance, suggesting that racial balancing can be restructured in ways that the Supreme Court may view as constitutional. Part III discusses more radical strategies that can be used to promote racial balance, and advocates a direct confrontation with the institution of judicial review in the context of affirmative action. The article concludes that the political branches of government possess the power to overcome Supreme Court impediments to racial justice, and hopes that they also possess the will to exercise that power

Just Do It

Racial injustice has always been a problem in the United States. The most salient victims of the Nation\u27s discrimination against racial minorities have included indigenous Indians, Chinese immigrants, Japanese-American citizens, Latinos, and of course blacks. But as the current war on terrorism illustrates, under the right conditions, almost any racial group can come within the scope of America\u27s discriminatory focus. It is common to suppose that that there is a difference between the progressive and the conservative ends of the political spectrum concerning the issue of race. However, those commonly accepted differences pale in comparison to the overriding similarity that exists between progressives and conservatives. Both progressives and conservatives are liberals in the classical sense of the term. And the tenets of liberalism seem destined to preclude us from ever achieving any meaningful level of racial equality in the United States