Geometric and Combinatorial Realizations of Crystal Graphs

For irreducible integrable highest weight modules of the finite and affine Lie algebras of type A and D, we define an isomorphism between the geometric realization of the crystal graphs in terms of irreducible components of Nakajima quiver varieties and the combinatorial realizations in terms of Young tableaux and Young walls. For type A_n⁽¹⁾, we extend the Young wall construction to arbitrary level, describing a combinatorial realization of the crystals in terms of new objects which we call Young pyramids. Presented by P. Littleman Mathematics Subject Classifications (2000) Primary 16G10, 17B37. Alistair Savage: This research was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada and was partially conducted by the author for the Clay Mathematics Institute.     

THE ENERGY ASPECT OF THE RECIPROCAL INTERACTIONS OF PAIRS OF TWO DIFFERENT VIBRATION MODES OF A CLAMPED ANNULAR PLATE

The standardized mutual active and reactive sound power of a clamped plate, representing the energy aspect of the reciprocal interactions of two different in vacuo modes, has been computed. It was assumed that the vibrations are axisymmetric, elastic and time harmonic, the plate's transverse deflection is small as compared with the plate's size, and that the vibration velocity is small as compared with the acoustic wavenumber generated. The Kirchhoff-Love theory of a perfectly elastic plate was used. The integral formulae for the mutual sound power were transformed into their Hankel representations which made possible their subsequent computation. A closed path integral was used to express the integral in its Hankel representation to compute the mutual active sound power. The asymptotic stationary phase method was used to compute the two magnitudes, i.e., the mutual active and reactive sound power. The results obtained are the asymptotic formulae valid for the acoustically fast waves. The oscillating as well as the non-oscillating terms have been identified in the formulae to make possible their further separate analysis. The availability of the asymptotic formulae makes possible some fast numerical computations of the mutual sound power. Moreover, the formulae presented herein, together with those for the individual modes known from the literature, make a complete basis for further computations of the total sound power of the plate's damped and forced vibrations in fluid.     

Detection of VX contamination in soil through solid-phase microextraction sampling and gas chromatography/mass spectrometry of the VX degradation product bis(diisopropylaminoethyl)disulfide

A solid-phase microextraction (SPME) and gas chromatography-mass spectrometry (GC-MS) sampling and analysis method was developed for bis(diisopropylaminoethyl)disulfide (a degradation product of the nerve agent VX) in soil. A 30-min sampling time with a polydimethylsiloxane-coated fiber and high temperature alkaline hydrolysis allowed detection with 1.0 microg of VX spiked per g of agricultural soil. The method was successfully used in the field with portable GC-MS instrumentation. This method is relatively rapid (less than 1 h), avoids the use of complex preparation steps, and enhances analyst safety through limited use of solvents and decontamination of the soil before sampling.     

Synthesis of trans‐disubstituted cyclam ligands appended with two 6‐hydroxymethylpyridin‐2‐ylmethyl sidearms: Crystal structures of the 1,8‐dimethyl‐4,ll‐di(6‐hydroxymethylpyridin‐2‐ylmethyl)cylam ligand and its Co(II) and Ni(II) complexes

Two new trans‐disubstituted cyclam ligands; 1,8‐di(6‐hydroxymethylpyridin‐2‐ylmethyl)‐1,4,8,11‐tetra‐azacyclotetradecane ( 5 ) and 1,8‐dimethyl‐4, 11‐di(6‐hydroxymethylpyridin‐2‐ylmethyl)‐1,4,8,11 ‐tetraaza‐cyclotetradecane ( 6 ); have been synthesized and characterized. The crystal structures of ligand 6 and its Ni(II) and Co(II) complexes have been determined. Crystal data are given for 6 , space group, P2₁/c, a = 11.095 (6) Å, b = 9.467 (5) Å, c = 13.283 (8) Å; β = 106.95 (5)°, Z = 2, R = 0.0715; for [Ni 6 ](C10₄)₂, space group P2₁/c, a = 9.4848 (14) Å, b = 33.941(6) Å, c = 9.793(2) A, β = 95.264(14)°, Z = 4, R = 0.0567; for [Co 6 ](C10₄)₂, space group, P2₁/c, a = 9.440 (6) Å, b = 33.848 (13) Å, c = 9.820 (3) Å, β = 95.16(3)°, Z = 4, R = 0.0718. In both complexes, the metal atoms are six‐coordinate with only one of the pendants interacting with the central metal atom and the other pendant remaining uncoordinated.