Lagrangean Structure and Propagation of Singularities in Multidimensional Compressible Flow

We study the propagation of singularities in solutions of the Navier–Stokes equations of compressible, barotropic fluid flow in two and three space dimensions. The solutions considered are in a fairly broad regularity class for which initial densities are nonnegative and essentially bounded, initial energies are small, and initial velocities are in certain fractional Sobolev spaces. We show that, if the initial density is bounded below away from zero in an open set V, then each point of V determines a unique integral curve of the velocity field and that this system of integral curves defines a locally bi-Hölder homeomorphism of V onto its image at each positive time. This “Lagrangean structure” is then applied to show that, if the initial density has a limit at a point of such a set V from a given side of a continuous hypersurface in V, then at each later time both the density and the divergence of the velocity have limits at the transported point from the corresponding side of the transported hypersurface, which is also a continuous manifold. If the limits from both sides exist, then the Rankine–Hugoniot conditions hold in a strict pointwise sense, showing that the jump in the divergence of the velocity is proportional to the jump in the pressure. This leads to a derivation of an explicit representation for the strength of the jump in the logarithm of the density, from which it follows that discontinuities persist for all time, convecting along fluid particle paths, and in the case that the pressure is strictly increasing in density, having strengths which decay exponentially in time.     

Strong convergence to global solutions for multidimensional flows of compressible,viscous fluids with polytropic equations of state and discontinuous initial data

We extend to general polytropic pressures P() = K, > 1, the existence theory of [8] for isothermal (= 1) flows of Navier-Stokes fluids in two and three space dimensions, with fairly general initial data. Specifically, we require that the initial density be close to a constant in L ² and L , and that the initial velocity be small in L ² and bounded in L ^{2 n} (in two dimensions the L ² norms must be weighted slightly). Solutions are obtained as limits of approximate solutions corresponding to mollified initial data. The key point is that the approximate densities are shown to converge strongly, so that nonlinear pressures can be accommodated, even in the absence of any uniform regularity information for the approximate densities.     

Waveform control of orientation-dependent ionization of DCl in few-cycle laser fields

Strong few-cycle light fields with stable electric field waveforms allow controlling electrons on time scales down to the attosecond domain. We have studied the dissociative ionization of randomly oriented DCl in 5 fs light fields at 720 nm in the tunneling regime. Momentum distributions of D(+) and Cl(+) fragments were recorded via velocity-map imaging. A waveform-dependent anti-correlated directional emission of D(+) and Cl(+) fragments is observed. Comparison of our results with calculations indicates that tailoring of the light field via the carrier envelope phase permits the control over the orientation of DCl(+) and in turn the directional emission of charged fragments upon the breakup of the molecular ion.     

Cause–effect analysis: improvement of a first year engineering students’ calculus teaching model

This study focuses on the mathematics department at a South African university and in particular on teaching of calculus to first year engineering students. The paper reports on a cause–effect analysis, often used for business improvement. The cause–effect analysis indicates that there are many factors that impact on secondary school teaching of mathematics, factors that the tertiary sector has no control over. The analysis also indicates the undesirable issues that are at the root of impeding success in the calculus module. Most important is that students are not encouraged to become independent thinkers from an early age. This triggers problems in follow-up courses where students are expected to have learned to deal with the work load and understanding of certain concepts. A new model was designed to lessen the impact of these undesirable issues.     

High-Sensitivity Immunofluorescence Staining: A Comparison of the Liposome Procedure and the FASER Technique on mGR Detection

Flow cytometry has become a widely-used and powerful tool for the characterization of cells according to their expression of specific proteins. However, sensitivity of this method is still limited since conventionally labeled antibodies can be conjugated with at maximum 1–10 dye molecules. This fact resulted in the need to develop new techniques in order to identify molecules which are expressed in very low but functionally relevant amounts. In the past, we have successfully used a liposome-based high-sensitivity immunofluorescence technique to measure the expression of low abundant membrane bound glucocorticoid receptors (mGR) on different cell types. The use of this technique allows the detection of as few as 50–100 antigen molecules per cell which is due to a 100-fold to 1000-fold increase in fluorescence signal intensity compared with conventional methods. The higher sensitivity is achieved since thousands of dye molecules can be enclosed in liposomes. Another modern high-sensitivity immunofluorescence staining method is the purchasable Fluorescence Amplification by Sequential Employment of Reagents (FASER) procedure. Here, we aimed at comparing sensitivity and specificity of these two techniques for the detection of the mGR. Our data demonstrate the FASER technique to be more sensitive and also more specific for the detection of mGR as compared to the liposome technique. However, both methods have advantages and disadvantages which are discussed in detail.     

On the role of organic adlayers in the anomalous water sorptivity of Lépine limestone

Sorptivity data are reported for the capillary absorption of water, ethanol, propan-2-ol, and n-heptane by the calcitic limestone Lépine (Lavoux à grain). The data confirm that the water sorptivity is anomalously low, an indication of partial wetting by water. Results are expressed in terms of a wetting index. The water sorptivity increases after heat treatment and chemical oxidation by hydrogen peroxide bleaching, while the sorptivity with organic liquids is unchanged. These treatments, therefore, increase the water wetting index. The results provide strong evidence that the presence of a natural organic adlayer is responsible for the anomalously low water sorptivity of this particular limestone. This natural water repellency effect may be exploited in developing chemical treatments to modify the water transport properties of stone.     

Thermodynamic and kinetic studies of H atom transfer from HMo(CO)3(eta(5)-C5H5) to Mo(N[t-Bu]Ar)3 and (PhCN)Mo(N[t-Bu]Ar)3: direct insertion of benzonitrile into the Mo-H bond of HMo(N[t-Bu]Ar)3 forming (Ph(H)C=N)Mo(N[t-Bu]Ar)3

Synthetic studies are reported that show that the reaction of either H2SnR2 (R = Ph, n-Bu) or HMo(CO)3(Cp) (1-H, Cp = eta(5)-C5H5) with Mo(N[t-Bu]Ar)3 (2, Ar = 3,5-C6H3Me2) produce HMo(N[t-Bu]Ar)3 (2-H). The benzonitrile adduct (PhCN)Mo(N[t-Bu]Ar)3 (2-NCPh) reacts rapidly with H2SnR2 or 1-H to produce the ketimide complex (Ph(H)C=N)Mo(N[t-Bu]Ar)3 (2-NC(H)Ph). The X-ray crystal structures of both 2-H and 2-NC(H)Ph are reported. The enthalpy of reaction of 1-H and 2 in toluene solution has been measured by solution calorimetry (DeltaH = -13.1 +/- 0.7 kcal mol(-1)) and used to estimate the Mo-H bond dissociation enthalpy (BDE) in 2-H as 62 kcal mol(-1). The enthalpy of reaction of 1-H and 2-NCPh in toluene solution was determined calorimetrically as DeltaH = -35.1 +/- 2.1 kcal mol(-1). This value combined with the enthalpy of hydrogenation of [Mo(CO)3(Cp)]2 (1(2)) gives an estimated value of 90 kcal mol(-1) for the BDE of the ketimide C-H of 2-NC(H)Ph. These data led to the prediction that formation of 2-NC(H)Ph via nitrile insertion into 2-H would be exothermic by approximately 36 kcal mol(-1), and this reaction was observed experimentally. Stopped flow kinetic studies of the rapid reaction of 1-H with 2-NCPh yielded DeltaH(double dagger) = 11.9 +/- 0.4 kcal mol(-1), DeltaS(double dagger) = -2.7 +/- 1.2 cal K(-1) mol(-1). Corresponding studies with DMo(CO)3(Cp) (1-D) showed a normal kinetic isotope effect with kH/kD approximately 1.6, DeltaH(double dagger) = 13.1 +/- 0.4 kcal mol(-1) and DeltaS(double dagger) = 1.1 +/- 1.6 cal K(-1) mol(-1). Spectroscopic studies of the much slower reaction of 1-H and 2 yielding 2-H and 1/2 1(2) showed generation of variable amounts of a complex proposed to be (Ar[t-Bu]N)3Mo-Mo(CO)3(Cp) (1-2). Complex 1-2 can also be formed in small equilibrium amounts by direct reaction of excess 2 and 1(2). The presence of 1-2 complicates the kinetic picture; however, in the presence of excess 2, the second-order rate constant for H atom transfer from 1-H has been measured: 0.09 +/- 0.01 M(-1) s(-1) at 1.3 degrees C and 0.26 +/- 0.04 M(-1) s(-1) at 17 degrees C. Study of the rate of reaction of 1-D yielded kH/kD = 1.00 +/- 0.05 consistent with an early transition state in which formation of the adduct (Ar[t-Bu]N)3Mo...HMo(CO)3(Cp) is rate limiting.     

Compressible Flow in a Half-Space with Navier Boundary Conditions

We prove the global existence of weak solutions of the Navier–Stokes equations of compressible flow in a half-space with the boundary condition proposed by Navier: the velocity on the boundary is proportional to the tangential component of the stress. This boundary condition allows for the determination of the scalar function in the Helmholtz decomposition of the acceleration density, which in turn is crucial in obtaining pointwise bounds for the density. Initial data and solutions are small in energy-norm with nonnegative densities having arbitrarily large sup-norm. These results generalize previous results for solutions in the whole space and are the first for solutions in this intermediate regularity class in a region with a boundary.