Unsteady oblique shock waves

A spatially and temporally local analysis is provided for unsteady, oblique shock waves, in which the flow is assumed to be two-dimensional or axisymmetric. Three unsteady parameters, in a laboratory frame, are viewed as the known independent variables. These are the upstream Mach number, the shock Mach number, and the angle of the shock relative to the instantaneous upstream velocity. Other steady and unsteady parameters, such as the velocity turn angles and downstream Mach numbers, are evaluated in closed form, in terms of these three quantities. Trends are assessed, and a sensitivity analysis is provided. It is suggested that the theory may find application in converting a shock capturing algorithm, at an early time during the computational process, into a shock fitting algorithm. Received 30 April 1999 / Accepted 29 November 1999     

On the role of convexity in isoperimetry,spectral gap and concentration

We show that for convex domains in Euclidean space, Cheeger’s isoperimetric inequality, spectral gap of the Neumann Laplacian, exponential concentration of Lipschitz functions, and the a-priori weakest requirement that Lipschitz functions have arbitrarily slow uniform tail-decay, are all quantitatively equivalent (to within universal constants, independent of the dimension). This substantially extends previous results of Maz’ya, Cheeger, Gromov–Milman, Buser and Ledoux. As an application, we conclude a sharp quantitative stability result for the spectral gap of convex domains under convex perturbations which preserve volume (up to constants) and under maps which are “on-average” Lipschitz. We also provide a new characterization (up to constants) of the spectral gap of a convex domain, as one over the square of the average distance from the “worst” subset having half the measure of the domain. In addition, we easily recover and extend many previously known lower bounds on the spectral gap of convex domains, due to Payne–Weinberger, Li–Yau, Kannan–Lovász–Simonovits, Bobkov and Sodin. The proof involves estimates on the diffusion semi-group following Bakry–Ledoux and a result from Riemannian Geometry on the concavity of the isoperimetric profile. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the CD(0,∞) curvature-dimension condition of Bakry-émery. Supported by NSF under agreement #DMS-0635607.     

On Gaussian Marginals of Uniformly Convex Bodies

Recently, Bo’az Klartag showed that arbitrary convex bodies have Gaussian marginals in most directions. We show that Klartag’s quantitative estimates may be improved for many uniformly convex bodies. These include uniformly convex bodies with power type 2, and power type p>2 with some additional type condition. In particular, our results apply to all unit-balls of subspaces of quotients of L _p for 1<p<∞. The same is true when L _p is replaced by S _p ^m , the l _p -Schatten class space. We also extend our results to arbitrary uniformly convex bodies with power type p, for 2≤p<4. These results are obtained by putting the bodies in (surprisingly) non-isotropic positions and by a new concentration of volume observation for uniformly convex bodies. Supported in part by BSF and ISF.     

Thermodynamic properties of quinoxaline-1,4-dioxide derivatives: a combined experimental and computational study

The mean (N-O) bond dissociation enthalpies were derived for three 2-methyl-3-(R)-quinoxaline 1,4-dioxide (1) derivatives, with R = methyl (1a), ethoxycarbonyl (1b), and benzyl (1c). The standard molar enthalpies of formation in the gaseous state at T = 298.15 K for the three 1 derivatives were determined from the enthalpies of combustion of the crystalline solids and their enthalpies of sublimation. In parallel, accurate density functional theory-based calculations were carried out in order to estimate the gas-phase enthalpies of formation for the corresponding quinoxaline derivatives. Also, theoretical calculations were used to obtain the first and second N-O dissociation enthalpies. These dissociation enthalpies are in excellent agreement with the experimental results herewith reported.     

Truncation of small matrix elements based on the Euclidean norm for blocked data structures

Methods for the removal of small symmetric matrix elements based on the Euclidean norm of the error matrix are presented in this article. In large scale Hartree-Fock and Kohn-Sham calculations it is important to be able to enforce matrix sparsity while keeping errors under control. Truncation based on some unitary-invariant norm allows for control of errors in the occupied subspace as described in (Rubensson et al. J Math Phys 49, 032103). The Euclidean norm is unitary-invariant and does not grow intrinsically with system size and is thus suitable for error control in large scale calculations. The presented truncation schemes repetitively use the Lanczos method to compute the Euclidean norms of the error matrix candidates. Ritz value convergence patterns are utilized to reduce the total number of Lanczos iterations.     

ACE applied to the quantitative characterization of benzo‐18‐crown‐6‐ether binding with alkali metal ions in a methanol–water solvent system

ACE was applied to the quantitative evaluation of noncovalent binding interactions between benzo‐18‐crown‐6‐ether (B18C6) and several alkali metal ions, Li⁺, Na⁺, K⁺, Rb⁺ and Cs⁺, in a mixed binary solvent system, methanol–water (50/50 v/v). The apparent binding (stability) constants (K_b) of B18C6–alkali metal ion complexes in the hydro‐organic medium above were determined from the dependence of the effective electrophoretic mobility of B18C6 on the concentration of alkali metal ions in the BGE using a nonlinear regression analysis. Before regression analysis, the mobilities measured by ACE at ambient temperature and variable ionic strength of the BGE were corrected by a new procedure to the reference temperature, 25°C, and the constant ionic strength, 10 mM . In the 50% v/v methanol–water solvent system, like in pure methanol, B18C6 formed the strongest complex with potassium ion (log K_b=2.89±0.17), the weakest complex with cesium ion (log K_b=2.04±0.20), and no complexation was observed between B18C6 and the lithium ion. In the mixed methanol–water solvent system, the binding constants of the complexes above were found to be about two orders lower than in methanol and about one order higher than in water.