The Equitable Basis for sl_2

This article contains an investigation of the equitable basis for the Lie algebra sl_2. Denoting this basis by {x,y,z}, we have [x,y] = 2x + 2y, [y,z] = 2y + 2z, [z, x] = 2z + 2x. One focus of our study is the group of automorphisms G generated by exp(ad x*), exp(ad y*), exp(ad z*), where {x*,y*,z*} is the basis for sl_2 dual to {x,y,z} with respect to the trace form (u,v) = tr(uv). We show that G is isomorphic to the modular group PSL_2(Z). Another focus of our investigation is the lattice L=Zx+Zy+Zz. We prove that the orbit G(x) equals {u in L |(u,u)=2}. We determine the precise relationship between (i) the group G, (ii) the group of automorphisms for sl_2 that preserve L, (iii) the group of automorphisms and antiautomorphisms for sl_2 that preserve L, and (iv) the group of isometries for (,) that preserve L. We obtain analogous results for the lattice L* =Zx*+Zy*+Zz*. Relative to the equitable basis, the matrix of the trace form is a Cartan matrix of hyperbolic type; consequently,we identify the equitable basis with the set of simple roots of the corresponding Kac-Moody Lie algebra g. Then L is the root lattice for g and 1/2L* is the weight lattice, and G(x) coincides with the set of real roots for g. Using L, L*, and G, we give several descriptions of the isotropic roots for g and show that each isotropic root has multiplicity 1. We describe the finite-dimensional sl_2-modules from the point of view of the equitable basis. In the final section, we establish a connection between the Weyl group orbit of the fundamental weights of g and Pythagorean triples.Comment: Minor changes made to the introductory material, and a few typos corrected. The final publication is available at http://www.springerlink.co

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Large internal waves in Massachusetts Bay transport sediments offshore

This paper is not subject to U.S. copyright. The definitive version was published in Continental Shelf Research 26 (2006): 2029-2049, doi:10.1016/j.csr.2006.07.022.A field experiment was carried out in Massachusetts Bay in August 1998 to assess the role of large-amplitude internal waves (LIWs) in resuspending bottom sediments. The field experiment consisted of a four-element moored array extending from just west of Stellwagen Bank (90-m water depth) across Stellwagen Basin (85- and 50-m water depth) to the coast (24-m water depth). The LIWs were observed in packets of 5–10 waves, had periods of 5–10 min and wavelengths of 200–400 m, and caused downward excursions of the thermocline of as much as 30 m. At the 85-m site, the current measured 1 m above bottom (mab) typically increased from near 0 to 0.2 m/s offshore in a few minutes upon arrival of the LIWs. At the 50-m site, the near-bottom offshore flow measured 6 mab increased from about 0.1 to 0.4–0.6 m/s upon arrival of the LIWs and remained offshore in the bottom layer for 1–2 h. The near-bottom currents associated with the LIWs, in concert with the tidal currents, were directed offshore and sufficient to resuspend the bottom sediments at both the 50- and 85-m sites. When LIWs are present, they may resuspend sediments for as long as 5 hours each tidal cycle as they travel westward across Stellwagen Basin. At 85-m water depth, resuspension associated with LIWs is estimated to occur for about 0.4 days each summer, about the same amount of time as caused by surface waves.MBIWE98 was supported by the USGS and the Office of Naval Research (ONR). The long-term observations at LT-A and LT-B were conducted under a Joint Funding Agreement between the USGS and the Massachusetts Water Resources Authority and an Inter-Service Agreement with the US Coast Guard. A. Scotti received support from the WHOI Postdoctoral Scholar program, the Johnson Foundation, the USGS, and ONR through grant N00014-01-1-0172; R. Beardsley through ONR grants N00014-98-1-0059, N00014-00-1-0210 and the WHOI Smith Chair in Coastal Physical Oceanography; and S. Anderson through ONR grant N000140-97-1-0158

Short-range order in multicomponent materials

The generalized multicomponent short-range order (GM-SRO) parameter has been adapted for the characterization of short-range order within the highly chemically and spatially resolved three-dimensional atomistic images provided by the microscopy technique of atom-probe tomography (APT). It is demonstrated that, despite the experimental limitations of APT, in many cases the GM-SRO results derived from APT data can provide a highly representative description of the atomic scale chemical arrangement in the original specimen. Further, based upon a target set of the GM-SRO parameters, measured from APT experiments, a Monte Carlo algorithm was utilized to simulate statistically equivalent atomistic systems which, unlike APT data, are complete and lattice based. The simulations replicate solute structures that are statistically consistent with other correlation measures such as solute cluster distributions, enable more quantitative characterization of nanostructural phenomena in the original specimen and, significantly, can be incorporated directly into other models and simulations