330 research outputs found

    Expressive Harms, Bizarre Districts, and Voting Rights: Evaluating Election-District Appearances After \u3cem\u3eShaw v. Reno\u3c/em\u3e

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    This article attempts to define the constitutional principles that characterize Shaw and to suggest how those principles might be applied in a consistent, meaningful way. Part I, in which we argue that Shaw must be understood to rest on a distinctive conception of the kinds of harms against which the Constitution protects, is the theoretical heart of the article. We call these expressive harms, as opposed to more familiar, material harms. In Part II, we briefly survey the history of previous, largely unsuccessful, efforts in other legal contexts to give principled content to these kinds of harms in redistricting. Parts III and IV then provide an alternative for evaluating district appearance by developing a quantitative approach for measuring district shapes that is most consistent with the theory of Shaw. These Parts are the empirical and social-scientific heart of the article. We apply our quantitative approach to congressional districts throughout the country, enabling meaningful comparisons between the congressional district at issue in Shaw and other districts. We also compare the shapes of congressional districts historically to test whether the district in Shaw is a distinctly recent phenomenon. In doing so, we identify the kind of districts most constitutionally vulnerable after Shaw. In Part V, we describe the further questions that lower courts must answer in deciding whether particular vulnerable districts ultimately fail the constitutional standard outlined in Shaw

    Phase Space Isometries and Equivariant Localization of Path Integrals in Two Dimensions

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    By considering the most general metric which can occur on a contractable two dimensional symplectic manifold, we find the most general Hamiltonians on a two dimensional phase space to which equivariant localization formulas for the associated path integrals can be applied. We show that in the case of a maximally symmetric phase space the only applicable Hamiltonians are essentially harmonic oscillators, while for non-homogeneous phase spaces the possibilities are more numerous but ambiguities in the path integrals occur. In the latter case we give general formulas for the Darboux Hamiltonians, as well as the Hamiltonians which result naturally from a generalized coherent state formulation of the quantum theory which shows that again the Hamiltonians so obtained are just generalized versions of harmonic oscillators. Our analysis and results describe the quantum geometry of some two dimensional systems.Comment: 26 pages, plain TeX; UBCTP 93-01

    Conformal Motions and the Duistermaat-Heckman Integration Formula

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    We derive a geometric integration formula for the partition function of a classical dynamical system and use it to show that corrections to the WKB approximation vanish for any Hamiltonian which generates conformal motions of some Riemannian geometry on the phase space. This generalizes previous cases where the Hamiltonian was taken as an isometry generator. We show that this conformal symmetry is similar to the usual formulations of the Duistermaat-Heckman integration formula in terms of a supersymmetric Ward identity for the dynamical system. We present an explicit example of a localizable Hamiltonian system in this context and use it to demonstrate how the dynamics of such systems differ from previous examples of the Duistermaat-Heckman theorem.Comment: 13 pages LaTeX, run twice. Uses epsf.tex, 2 postscript files read directly into LaTeX file from director
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