Racah Polynomials and Recoupling Schemes of su(1,1)\mathfrak{su}(1,1)

The connection between the recoupling scheme of four copies of $\mathfrak{su}(1,1)$, the generic superintegrable system on the 3 sphere, and bivariate Racah polynomials is identified. The Racah polynomials are presented as connection coefficients between eigenfunctions separated in different spherical coordinate systems and equivalently as different irreducible decompositions of the tensor product representations. As a consequence of the model, an extension of the quadratic algebra ${\rm QR}(3)$ is given. It is shown that this algebra closes only with the inclusion of an additional shift operator, beyond the eigenvalue operators for the bivariate Racah polynomials, whose polynomial eigenfunctions are determined. The duality between the variables and the degrees, and hence the bispectrality of the polynomials, is interpreted in terms of expansion coefficients of the separated solutions

Periodic Manifolds with Spectral Gaps

We investigate spectral properties of the Laplace operator on a class of non-compact Riemannian manifolds. For a given number $N$ we construct periodic (i.e. covering) manifolds such that the essential spectrum of the corresponding Laplacian has at least $N$ open gaps. We use two different methods. First, we construct a periodic manifold starting from an infinite number of copies of a compact manifold, connected by small cylinders. In the second construction we begin with a periodic manifold which will be conformally deformed. In both constructions, a decoupling of the different period cells is responsible for the gaps.Comment: 21 pages, 3 eps-figures, LaTe

Note on 'N-pseudoreductions' of the KP hierarchy

The group-theoretical side of N-pseudoreductions is discussed. The resulting equations are shown to be easy transformations of the N-KdV hierarch

On the structure of graded transitive Lie algebras

We study finite-dimensional Lie algebras ${\mathfrak L}$ of polynomial vector fields in $n$ variables that contain the vector fields $\dfrac{\partial}{\partial x_i} \; (i=1,\ldots, n)$ and $x_1\dfrac{\partial}{\partial x_1}+ \dots + x_n\dfrac{\partial}{\partial x_n}$. We show that the maximal ones always contain a semi-simple subalgebra $\bar{{\mathfrak g}}$, such that $\dfrac{\partial}{\partial x_i}\in \bar{{\mathfrak g}} \; (i=1,\ldots, m)$ for an $m$ with $1 \leq m \leq n$. Moreover a maximal algebra has no trivial $\bar{{\mathfrak g}}$-module in the space spanned by $\dfrac{\partial}{\partial x_i} (i=m+1,\ldots, n)$. The possible algebras $\bar{{\mathfrak g}}$ are described in detail, as well as all $\bar{{\mathfrak g}}$-modules that constitute such maximal ${\mathfrak L}$. All maximal ${\mathfrak L}$ are described explicitly for $n\leq 3$

Marshall as a Judge

Marshall is a towering and inspirational figure in the history of American constitutional law. He changed American life forever and unquestionably for the better. But the contemporary significance of Marshall’s legacy is also, in ways that challenge present practices and beliefs, ambiguous

Federalism in the Taft Court Era: Can It Be “Revived”?

This Article analyzes the Supreme Court\u27s view of federalism during the decade of the 1920s. It offers a detailed discussion of four jurisprudential areas: congressional power, dormant Commerce Clause doctrine, intergovernmental tax immunity, and judicial centralization through the enforcement of federal common law and constitutional rights. The resurgent federalism of the contemporary Court is typically characterized as reviving pre-New Deal principles. This Article concludes, however, that any such revival is highly implausible. It offers four reasons for this conclusion. First, the pre-New Deal Court conceived federalism in terms of the ideal of dual sovereignty, which imagined that the federal government and the states regulated distinct and exclusive spheres of social and economic life. But because the national market had by the twentieth century become thoroughly integrated, this ideal produced doctrinal incoherence in the areas of both intergovernmental tax immunity and the dormant Commerce Clause. The application of the ideal of dual sovereignty also significantly undercut state power, because it invited the pre-New Deal Court to prohibit states from regulating the exclusively federal area of interstate commerce. For these reasons the modern Court has abandoned the ideal of dual sovereignty in its doctrine of intergovernmental tax immunity and the dormant Commerce Clause. Contemporary opinions in these areas imagine federal and state interests as intermingled and overlapping, rather than as separated into discrete spheres. The modern view actually offers more protection for state regulations than did the ideal of dual sovereignty espoused by the pre-New Deal Court. Second, the pre-New Deal Court understood itself as a common law court authorized to articulate the deepest experiences and values of the American people. This authority transcended the distinction between federal and state power, which is why the pre-New Deal Court never conceived itself as an agent of a federal government that was potentially in tension with state sovereignty. The Court never understood the centralization resulting from judicial decisionmaking as a federalism issue. The Court freely regulated intimate areas of state life through the promulgation of general common law. The pre-New Deal Court\u27s common law authority was regarded as even more fundamental than Congress\u27s claim to articulate the national will. The triumph of Holmesian positivism in Erie Railroad Co. v. Tompkins transformed the Court into an instrument of specifically federal law. The federalism implications of judicial decisionmaking in the areas of common law and constitutional rights were thus made manifest for the first time. The Court\u27s authority to impose structural limitations on congressional power was also profoundly altered. Third, the pre-New Deal Court, like the country generally, regarded the federal government as a potentially distant, bureaucratic, and oppressive institution. States were by contrast conceptualized as sites of democratic self-government. Federalism was typically conceived as the problem of reconciling centralization with self-government. Thus federal and state regulations, even of the same subject matter, were not regarded as equivalent. State regulation was self-chosen; federal regulation was potentially coercive. This view of the federal government was pushed to the margins of American political culture when the crisis of the New Deal legitimated the national government\u27s authority to speak as the genuine representative of an authentic national democratic will. Combined with the demise of the Court\u27s common law authority, this transformation of Congress\u27s legitimacy undercut the Court\u27s ability to second-guess Congress\u27s vision of national priorities when reviewing the limits of congressional power. Fourth, the pre-New Deal Court conceived structure and rights as complementary and mutually dependent concepts. The Court defined individual rights in ways designed to serve structural principles, like the integration of the national market. And it defined structural principles, like the limits of congressional power, in terms of the individual rights affected by federal legislation. Because the Lochnerism of the pre-New Deal Court inclined it to protect freedom of contract, it sought to impose limits on congressional power that were highly sensitive to the nature of the economic transactions regulated by federal legislation. Modern constitutional thinking, by contrast, sharply distinguishes structure from rights, and it does not seek to protect the same kind of economic rights as did pre-New Deal Lochnerism. The revival of pre-New Deal federalism, in short, would require the contemporary Court to restore an ideal of dual sovereignty that in important doctrinal areas is not only incoherent, but deeply antagonistic to state power; to reassert its authority as a common law court; to resurrect an image of Congress as a national legislature unsupported by a genuine national democratic will; and to dismantle the contemporary distinction between structure and rights so as to limit congressional power in ways designed to protect rights of substantive due process