An example of stability of singular spectrum under smooth perturbations

Random Fourier series with Gaussian coefficients and very large gaps are used to construct a class of continuous stationary Gaussian processes whose sample functions X(t) have remarkable properties. Almost surely the function X(t) has an essential range of Lebesgue measure zero and for each smooth function Y(t) the multiplication operator (M_x+yf) (t) [X(t)+Y(t)]f(t) on L²[0,2] has only singular continuous spectrum.     

Measuring particle size distribution using LED-illumination

A particle/droplet image analysis (PDIA) system employing LED-illumination was designed. Freezing the moving droplets using high speed camera instead of stroboscopic illumination, the system had no requirement to synchronize the backlight and the camera. It featured low cost, low power consumption and simple optical configuration in comparison to laser-based systems. Only focused droplets in images were counted. Given the sample size of the system was relatively small, an upper-limit lognormal distribution was used to fit the actual data to represent the spray patterns. The accuracy of the diameter measurement was verified using precisely manufactured balls. Comparisons with two laser-based systems were given and the system’s capacity to distinct spray patterns were demonstrated in nozzle classification experiments.     

Do Mathematical Equations Display Social Attributes?

This column is a forum for discussion of mathematical communities throughout the world, and through all time. Our definition of “mathematical community” is the broadest. We include “schools” of mathematics, circles of correspondence mathematical societies, student organizations, and informal communities of cardinality greater than one. What we say about the communities is just as unrestricted. We welcome contributions from mathematicians of all kinds and in all places, and also from scientists, historians, anthropologists, and others.     

Measurement of tobramycin by reversed-phase high-performance liquid chromatography with mass spectrometry detection

Analysis of tobramycin faces challenges owing to its significant basicity, hydrophilicity and lack of a UV absorbing chromophore. Chromatographic methods, coupled with derivatization to introduce chromophores for tobramycin analysis, were extensively studied. A direct reversed-phase HPLC method for tobramycin analysis has not been reported. Here, we would like to report a simple LC/MS method for quantitative analysis of tobramycin in pharmaceutical formulations. Reversed-phase HPLC analysis of tobramycin was achieved using a pH stable C18 column with basic (pH 11) aqueous mobile phase (ammonium hydroxide buffer), while direct detection was carried out employing a single quadruple mass detector in negative mode via electrospray ionization. This unique separation-detection combination provided simple and specific determination of tobramycin. This method was found to be linear at a tobramycin concentration range of 0.2-0.8 mg/mL with a correlation coefficient value of 0.999. The quantitation limit and detection limit were calculated as 0.210 and 0.063 μg/mL, respectively, with 99.994% confidence. This method was successfully applied to measure tobramycin content in matrices containing tobramycin and other pharmaceutical formulation ingredients. Recoveries of 101.8, 97.8 and 106.7% were obtained for tobramycin spiked in the pharmaceutical formulation at concentrations of 1.68, 1.0 and 0.35 mg/mL, respectively. The relative standard deviations for six injections of spiked samples ranged from 0.2 to 3.2%, indicating good method repeatability.     

Nondispersive two-electron wave packets in a helium atom

We demonstrate the existence of stable nondispersing two-electron wave packets in the helium atom in combined magnetic and circularly polarized microwave fields. These packets follow circular orbits and we show that they can also exist in quantum dots. Classically the two electrons follow trajectories which resemble orbits discovered by Langmuir and which were used in attempts at a Bohr-like quantization of the helium atom. Eigenvalues of a generalized Hessian matrix are computed to investigate the classical stability of these states. Diffusion Monte Carlo simulations demonstrate the quantum stability of these two-electron wave packets in the helium atom and quantum-dot helium with an impurity center.     

Observation of collapse of pseudospin order in bilayer quantum Hall ferromagnets

The Hartree-Fock paradigm of bilayer quantum Hall states with finite tunneling at filling factor nu=1 has full pseudospin ferromagnetic order with all the electrons in the lowest symmetric Landau level. Inelastic light scattering measurements of low energy spin excitations reveal major departures from the paradigm at relatively large tunneling gaps. The results indicate the emergence of a novel correlated quantum Hall state at nu=1 characterized by reduced pseudospin order. Marked anomalies occur in spin excitations when pseudospin polarization collapses by application of in-plane magnetic fields.     

Flow microcalorimetry of adsorption of anionic alkyl surfactants at the solid/liquid interface

Flow microcalorimetry was used to study the adsoption of anionic alkyl surfactants from aque--ous solutions onto silica. It is found that for alkyl sulfate systems the strength of adsorption interactionincreases with increases of the alkyl chain length and decreases as temperature rises. The adsorptiondepends only on monomer concentration of the solution even above the critical micelle concentration(cmc). The assumption is made that the adsorption involves only a transfer of monomers from bulkto surface phase. A different adsorption mechanism is operative for the alkyl carboxylate.     

Mobility of positrons in silicon

The drift velocity v₊ of positrons in Si has been measured by observing the Doppler shift of the annihilation γ's. The electric field dependence of v₊ yields the positron mobility μ₊: at 80 K μ₊=460±20 cm²V^-1 sec^-1 and at 184 K μ₊=173±15 cm²V^-1 sec^-1.     

Hasse invariants and anomalous primes for elliptic curves with complex multiplication

Let C be an elliptic curve defined over Q. Let p be a prime where C has good reduction. By definition, p is anomalous for C if the Hasse invariant at p is congruent to 1 modulo p. The phenomenon of anomalous primes has been shown by Mazur to be of great interest in the study of rational points in towers of number fields. This paper is devoted to discussing the Hasse invariants and the anomalous primes of elliptic curves admitting complex multiplication. The two special cases Y² = X³ + a₄X and Y² = X³ + a₆ are studied at considerable length. As corollaries, some results in elementary number theory concerning the residue classes of the binomial coefficients (_n²ⁿ) (Resp. (_n³ⁿ)) modulo a prime p = 4n + 1 (resp. p = 6n + 1) are obtained. It is shown that certain classes of elliptic curves admitting complex multiplication do not have any anomalous primes and that others admit only very few anomalous primes.