The Loewner Equation: Maps and Shapes

An approach called Schramm–Loewner evolution (SLE) provides a new method for dealing with a wide variety of scale-invariant problems in two dimensions. This approach is based upon an older method called Loewner Evolution (LE), which connects analytic and geometrical constructions in the complex plane. In this paper, the bases of LE and SLE are described and some simple applications are discussed in relatively non-technical form. A bibliography of the subject is presented.     

Nonlinear vibrations of a charged drop in a third-order approximation in the amplitude of multimode initial deformation

A solution to the problem of nonlinear surface vibration of a charged ideal liquid drop is found in a third-order approximation in initial multimode deformation of the equilibrium spherical shape by the method of many scales. It is shown that the spectrum of modes that are responsible for the shape of the drop at an arbitrary time instant depends considerably on the spectrum of modes governing the initial deformation of the drop. The latter spectrum also has an effect on nonlinear corrections to the vibration frequencies and, consequently, on a nonlinear correction to the critical Rayleigh parameter, which specifies the stability of the drop against self-charge.     

On numerical ranges and roots

Existence of the fractional powers is established in Banach algebra setting, in terms of the numerical ranges of elements involved. The behavior of the spectra and (for Hermitian ∗-algebras satisfying some additional hypotheses) the ∗-numerical range under taking these powers also is investigated.     

Optical bistability of planar metal/dielectric nonlinear nanostructures

We study theoretically a nonlinear response of the planar metal/dielectric nanostructures constituted from periodical array of ultra thin silver layers and the layers of Kerr-like nonlinear dielectric. We predict hysteresis-type dependences of the components of the tensor of effective dielectric permittivity on the field intensity allowing the change in material transmission properties from transparent to opaque and back at extremely low intensities of the light. It makes possible to control the light by light in all-optical nanoscale devices and circuits.     

Magnetic phase transitions in nanostructures induced by shear stress under high pressure

Magnetic phase transitions of the first and second order were revealed by Mössbauer spectroscopy in nanosystems of - and -ferric oxides and metallic europium subjected to shear stress (240°) under high pressure (20 kbar). For - and -ferric oxide nanoclusters, the Curie (Neel) points decreased to 300 K, whereas for nanostructured europium the Neel point increased from 90 to 100 K. The thermodynamic model of magnetic phase transitions predicting a change in the character of magnetic phase transitions and a decrease (increase) in the critical Neel (Curie) points in nanoclusters was developed. The type of magnetic phase transitions and the change in the critical points were caused by defects in nanoclusters, whose maximum concentration was observed for the clusters with the 20—50 nm size range.     

Myoglobin-CO substate structures and dynamics: multidimensional vibrational echoes and molecular dynamics simulations

Spectrally resolved infrared stimulated vibrational echo data were obtained for sperm whale carbonmonoxymyoglobin (MbCO) at 300 K. The measured dephasing dynamics of the CO ligand are in agreement with dephasing dynamics calculated with molecular dynamics (MD) simulations for MbCO with the residue histidine-64 (His64) having its imidazole epsilon nitrogen protonated (N(epsilon)-H). The two conformational substate structures B(epsilon) and R(epsilon) observed in the MD simulations are assigned to the spectroscopic A(1) and A(3) conformational substates of MbCO, respectively, based on the agreement between the experimentally measured and calculated dephasing dynamics for these substates. In the A(1) substate, the N(epsilon)-H proton and N(delta) of His64 are approximately equidistant from the CO ligand, while in the A(3) substate, the N(epsilon)-H of His64 is oriented toward the CO, and the N(delta) is on the surface of the protein. The MD simulations show that dynamics of His64 represent the major source of vibrational dephasing of the CO ligand in the A(3) state on both femtosecond and picosecond time scales. Dephasing in the A(1) state is controlled by His64 on femtosecond time scales, and by the rest of the protein and the water solvent on longer time scales.     

Comparability of analytical results obtained in proficiency testing based on a metrological approach

A “yes–no” type of criterion is proposed for the assessment of comparability of proficiency testing (PT) results when the PT scheme is based on a metrological approach, i.e. on the use of a reference material as the test sample, etc. The criterion tests a null hypothesis concerning the insignificance of a bias of the mean of the results from a traceable value certified in the reference material used for the PT. Reliability of such assessment is determined by the probabilities of not rejecting the null hypothesis when it is true, and rejecting it when it is false (the alternative hypothesis is true). It is shown that a number of chemical, metrological and statistical reasons should be taken into account for careful formulation of the hypotheses, enabling the avoidance of an erroneous assessment of the comparability. The criterion can be helpful for PT providers and laboratory accreditation bodies in analysis of PT results.