Thermal diffusion shock waves

The Ludwig-Soret effect or thermal diffusion, which refers to the separation of liquid mixtures in a temperature gradient, is governed by a nonlinear, partial differential equation in space and time. It is shown here that the solution to the nonlinear differential equation for a binary mixture predicts the existence of shock waves completely analogous to fluid shocks and obeys an expression for the shock velocity that is an exact analogue of the Rankine-Hugoniot relations. Direct measurements of the time dependent, spatial absorption profile of a suspension of nanometer sized particles subjected to a sinusoidal temperature field generated by a pair of continuous laser beams, as well as self-diffraction experiments, show motion of the particles in agreement with the predictions of nonlinear theory.     

A hybrid active/passive exhaust noise control system for locomotives

A prototype hybrid system consisting of active and passive components for controlling far-field locomotive exhaust noise has been designed, assembled, and tested on a locomotive. The system consisted of a resistive passive silencer for controlling high-frequency broadband noise and a feedforward multiple-input, multiple-output active control system for suppressing low-frequency tonal noise. The active system used ten roof-mounted bandpass speaker enclosures with 2-12-in. speakers per enclosure as actuators, eight roof-mounted electret microphones as residual sensors, and an optical tachometer that sensed locomotive engine speed as a reference sensor. The system was installed on a passenger locomotive and tested in an operating rail yard. Details of the system are described and the near-field and far-field noise reductions are compared against the design goal.     

Order ideals and a generalized Krull height theorem

Let N be a finitely generated module over a Noetherian local ring (R,m). We give criteria for the height of the order ideal N*(x) of an element x N to be bounded by the rank of N. The Generalized Principal Ideal Theorem of Bruns, Eisenbud and Evans says that this inequality always holds if x mN. We show that the inequality even holds if the hypothesis becomes true after first extending scalars to some local domain and then factoring out torsion. We give other conditions in terms of residual intersections and integral closures of modules. We derive information about order ideals that leads to bounds on the heights of trace ideals of modules—even in circumstances where we do not have the expected bounds for the heights of the order ideals!All three authors were partially supported by the NSF.Revised version: 24 November 2003     

Synthesis of peptide-nanotube platinum-nanoparticle composites

Nanotubes prepared by the self-assembly of D-Phe-D-Phe molecules are investigated by electron microscopy and Monte Carlo simulations; the nanotubes appear to be porous and are capable of forming novel peptide-nanotube platinum-nanoparticle composites.     

Differential Equations for Dyson Processes

We call a Dyson process any process on ensembles of matrices in which the entries undergo diffusion. We are interested in the distribution of the eigenvalues (or singular values) of such matrices. In the original Dyson process it was the ensemble of n×n Hermitian matrices, and the eigenvalues describe n curves. Given sets X₁,...,X_m the probability that for each k no curve passes through X_k at time _k is given by the Fredholm determinant of a certain matrix kernel, the extended Hermite kernel. For this reason we call this Dyson process the Hermite process. Similarly, when the entries of a complex matrix undergo diffusion we call the evolution of its singular values the Laguerre process, for which there is a corresponding extended Laguerre kernel. Scaling the Hermite process at the edge leads to the Airy process (which was introduced by Prähofer and Spohn as the limiting stationary process for a polynuclear growth model) and in the bulk to the sine process; scaling the Laguerre process at the edge leads to the Bessel process.In earlier work the authors found a system of ordinary differential equations with independent variable whose solution determined the probabilitieswhere A() denotes the top curve of the Airy process. Our first result is a generalization and strengthening of this. We assume that each X_k is a finite union of intervals and find a system of partial differential equations, with the end-points of the intervals of the X_k as independent variables, whose solution determines the probability that for each k no curve passes through X_k at time _k. Then we find the analogous systems for the Hermite process (which is more complicated) and also for the sine process. Finally we find an analogous system of PDEs for the Bessel process, which is the most difficult.Dedicated to Freeman Dyson on the occasion of his eightieth birthdayAcknowledgement We thank Kurt Johansson for sending us his unpublished notes on the extended Hermite kernel. This work was supported by National Science Foundation under grants DMS-0304414 (first author) and DMS-0243982 (second author).     

Adsorption pattern of mixtures of trimethylammonium-modified hydroxyethylcellulose and sodium dodecyl sulfate at solid-liquid interfaces

We studied mixtures of aqueous solutions of cationic hydroxyethylcellulose JR400 polymer and anionic sodium dodecyl sulfate using dynamic light scattering and atomic force microscopy (AFM). A ternary phase diagram was established showing three interesting realms of the polymer-surfactant-water mixture: a preprecipitation area of lowered viscosity (polymer excess) compared to the pure polymer solution, a postprecipitation area (resolubilization at surfactant excess), and highly diluted samples with a stoichiometrical surfactant-polymer ratio close to that of maximum precipitation. Samples with various compositions representing these areas were imaged by atomic force microscopy on mica and on hydrophobically modified silica in contact mode. A correlation between light scattering data concerning particle size and, more important, structuring in the bulk on one hand and AFM images on the other hand was observed. It was revealed that the influence of surface properties is of less importance for adsorption, compared to the influence of the mixture in the bulk, provided that the mixture is prepared prior to adsorption.     

Heavy alkaline earth metal pyrazolates: synthetic pathways, structural trends, and comparison with divalent lanthanoids

Two series of heavy alkaline earth metal pyrazolates, [M(Ph(2)pz)(2)(thf)(4)] 1 a-c (Ph(2)pz=3,5-diphenylpyrazolate, M=Ca, Sr, Ba; THF=tetrahydrofuran) and [M(Ph(2)pz)(2)(dme)(n)] (M=Ca, 2 a, Sr, 2 b, n=2; M=Ba, 2 c, n=3; DME=1,2-dimethoxyethane) have been prepared by redox transmetallation/ligand exchange utilizing Hg(C(6)F(5))(2). Compounds 1 a and 2 b were also obtained by redox transmetallation with Tl(Ph(2)pz). Alternatively, direct reaction of the alkaline earth metals with 3,5-diphenylpyrazole at elevated temperatures under solventless conditions yielded compounds 1 a-c and 2 a-c upon extraction with THF or DME. By contrast, [M(Me(2)pz)(2)(Me(2)pzH)(4)] 3 a-c (M=Ca, Sr, Ba; Me(2)pzH=3,5-dimethylpyrazole) were prepared by protolysis of [M[N(SiMe(3))(2)](2)(thf)(2)] (M=Ca, Sr, Ba) with Me(2)pzH in THF and by direct metallation with Me(2)pzH in liquid NH(3)/THF. Compounds 1 a-c and 2 a-c display eta(2)-bonded pyrazolate ligands, while 3 a,b exhibit eta(1)-coordination. Complexes 1 a-c have transoid Ph(2)pz ligands and an overall coordination number of eight with a switch from mutually coplanar Ph(2)pz ligands in 1 a,b to perpendicular in 1 c. In eight coordinate 2 a,b the pyrazolate ligands are cisoid, whilst 2 c has an additional DME ligand and a metal coordination number of ten. By contrast, 3 a,b have octahedral geometry with four eta(1)-Me(2)pzH donors, which are hydrogen-bonded to the uncoordinated nitrogen atoms of the two trans Me(2)pz ligands. The application of synthetic routes initially developed for the preparation of lanthanoid pyrazolates provides detailed insight into the similarities and differences between the two groups of metals and structures of their complexes.     

Unprecedented photochemical induced cascading rearrangement of the 3-azabicyclo[3.3.1]nonane skeleton

Certain 3-azabicyclo[3.3.1]nonane derivatives undergo unprecedented stereospecific skeletal cleavage when subjected to light affording a novel heterotricyclic skeleton.