Dimensions of Equality: Doctrines of Limitation

America can be justifiably proud of the enormous strides its legal system has made since the end of World War II in identifying and vindicating equality rights under the U.S. Constitution. The 1954 decision in Brown v. Board of Education, striking down the separate-but-equal doctrine in public education, provided the inspiration and the doctrinal basis for racial minorities, women, aliens, children born out of wedlock, the disabled, and the poor to pursue their claims for evenhanded treatment in the courts. We also have seen greater judicial protection of equality in the exercise of rights guaranteed by the first amendment to the Constitution, such as freedom of religion, speech, and the press. For all this progress, however, the Supreme Court has over the last two decades embraced doctrines of limitation that severely constrain the ability of equality claims to get a judicial hearing and to receive vindication. These doctrines raise serious questions as to whether the federal court system can be looked to in the future for meaningful protection of equality rights. It is to a brief discussion of a few of these doctrines-state action, discriminatory intent, and federalism-that I would like to turn

Sweep maps: A continuous family of sorting algorithms

We define a family of maps on lattice paths, called sweep maps, that assign levels to each step in the path and sort steps according to their level. Surprisingly, although sweep maps act by sorting, they appear to be bijective in general. The sweep maps give concise combinatorial formulas for the q,t-Catalan numbers, the higher q,t-Catalan numbers, the q,t-square numbers, and many more general polynomials connected to the nabla operator and rational Catalan combinatorics. We prove that many algorithms that have appeared in the literature (including maps studied by Andrews, Egge, Gorsky, Haglund, Hanusa, Jones, Killpatrick, Krattenthaler, Kremer, Orsina, Mazin, Papi, Vaille, and the present authors) are all special cases of the sweep maps or their inverses. The sweep maps provide a very simple unifying framework for understanding all of these algorithms. We explain how inversion of the sweep map (which is an open problem in general) can be solved in known special cases by finding a "bounce path" for the lattice paths under consideration. We also define a generalized sweep map acting on words over arbitrary alphabets with arbitrary weights, which is also conjectured to be bijective.Comment: 21 pages; full version of FPSAC 2014 extended abstrac

Momentum flow in black-hole binaries. I. Post-Newtonian analysis of the inspiral and spin-induced bobbing

A brief overview is presented of a new Caltech/Cornell research program that is exploring the nonlinear dynamics of curved spacetime in binary black-hole collisions and mergers, and of an initial project in this program aimed at elucidating the flow of linear momentum in binary black holes (BBHs). The “gauge-dependence” (arbitrariness) in the localization of linear momentum in BBHs is discussed, along with the hope that the qualitative behavior of linear momentum will be gauge-independent. Harmonic coordinates are suggested as a possibly preferred foundation for fixing the gauge associated with linear momentum. For a BBH or other compact binary, the Landau-Lifshitz formalism is used to define the momenta of the binary’s individual bodies in terms of integrals over the bodies’ surfaces or interiors, and define the momentum of the gravitational field (spacetime curvature) outside the bodies as a volume integral over the field’s momentum density. These definitions will be used in subsequent papers that explore the internal nonlinear dynamics of BBHs via numerical relativity. This formalism is then used, in the 1.5 post-Newtonian approximation, to explore momentum flow between a binary’s bodies and its gravitational field during the binary’s orbital inspiral. Special attention is paid to momentum flow and conservation associated with synchronous spin-induced bobbing of the black holes, in the so-called “extreme-kick configuration” (where two identical black holes have their spins lying in their orbital plane and antialigned)

Proposal for a Topological Plasmon Spin Rectifier

We propose a device in which the spin-polarized AC plasmon mode in the surface state of a topological insulator nanostructure induces a static spin accumulation in a resonant, normal metal structure coupled to it. Using a finite-difference time-domain model, we simulate this spin-pump mechanism with drift, diffusion, relaxation, and precession in a magnetic field. This optically-driven system can serve as a DC "spin battery" for spintronic devices.Comment: Eq. 1 corrected; Figs 3 and 4 update

Statistical deconvolution of enthalpic energetic contributions to MHC-peptide binding affinity

Background: MHC Class I molecules present antigenic peptides to cytotoxic T cells, which forms an integral part of the adaptive immune response. Peptides are bound within a groove formed by the MHC heavy chain. Previous approaches to MHC Class I-peptide binding prediction have largely concentrated on the peptide anchor residues located at the P2 and C-terminus positions. Results: A large dataset comprising MHC-peptide structural complexes was created by re-modelling pre-determined x-ray crystallographic structures. Static energetic analysis, following energy minimisation, was performed on the dataset in order to characterise interactions between bound peptides and the MHC Class I molecule, partitioning the interactions within the groove into van der Waals, electrostatic and total non-bonded energy contributions. Conclusion: The QSAR techniques of Genetic Function Approximation (GFA) and Genetic Partial Least Squares (G/PLS) algorithms were used to identify key interactions between the two molecules by comparing the calculated energy values with experimentally-determined BL50 data. Although the peptide termini binding interactions help ensure the stability of the MHC Class I-peptide complex, the central region of the peptide is also important in defining the specificity of the interaction. As thermodynamic studies indicate that peptide association and dissociation may be driven entropically, it may be necessary to incorporate entropic contributions into future calculations