Abstract:
This work studies the canonical representations (Berezin representations) for para-Hermitian symmetric spaces of rank one. These spaces are exhausted up to the covering by spaces G/H with G = SL(n,R),H = GL(n-1,R) . For Hermitian symmetric spaces G/K, canonical representations were introduced by Berezin and Vershik-Gelfand-Graev. They are unitary with respect to some invariant non-local inner product (the Berezin form). We consider canonical representations in a wider sense: we give up the condition of unitarity and let these representations act on spaces of distributions. For our spaces G/H, the canonical representations turn out to be tensor products of representations of maximal degenerate series and contragredient representations. We decompose the canonical representations into irreducible constituents and decompose boundary representations.

Abstract:
It is shown that the Josephson subsystem of the Lawrence-Doniach model of layered superconductors in the London approximation can be presented as a system with variable number of classical Coulomb particles. This allows us to consider the vortex system of a coupled layered superconductor as the system of these particles and 2D-vortices interacting with each other. The grand partition function of the system was written and transformed into the form of field one. Thermodynamical properties of the model obtained was studied. It is found that there is no a phase transition in the system. Instead of this the model demonstrates the crossover from a low temperature 3D behavior to high temperature 2D one which can look as a phase transition for experimental purposes.

Abstract:
The dynamics of short 1D nonlinear Hamiltonian chains is analyzed numerically at different temperatures (energy per particle). The boundary temperature $T_b$ separating the regular (quasiperiodic) and the stochastic (chaotic) chain motion is found. The dynamical properties of short 1D nonlinear chains interacting with thermostats are studied. It is shown that, in spite of the fluctuations, the dynamics of such systems can be stochastic as well as regular. The boundary temperature of these systems is close to that of the Hamiltonian one.

Abstract:
Dielectric spectra (10^4-10^11 Hz) of water and ice at 0 {\deg}C are considered in terms of proton conductivity and compared to each other. In this picture, the Debye relaxations, centered at 1/{\tau}_W ~ 20 GHz (in water) and 1/{\tau}_I ~ 5 kHz (in ice), are seen as manifestations of diffusion of separated charges in the form of H3O+ and OH- ions. The charge separation results from the self-dissociation of H2O molecules, and is accompanied by recombination in order to maintain the equilibrium concentration, N. The charge recombination is a diffusion-controlled process with characteristic lifetimes of {\tau}_W and {\tau}_I, for water and ice respectively. The static permittivity, {\epsilon}(0), is solely determined by N. Both, N and {\epsilon}(0), are roughly constant at the water-ice phase transition, and both increase, due to a slowing down of the diffusion rate, as the temperature is lowered. The transformation of the broadband dielectric spectra at 0 {\deg}C with the drastic change from {\tau}_W to {\tau}_I is mainly due to an abrupt (by 0.4 eV) change of the activation energy of the charge diffusion.

Abstract:
The work is devoted to the critical analysis of theoretical prediction and astronomical observation of GR effects, first of all, the Mercury's perihelion advance. In the first part, the methodological issues of observations are discussed including a practice of observations, a method of recognizing the relativistic properties of the effect and recovering it from bulk of raw data, a parametric observational model, and finally, methods of assessment of the effect value and statistical level of confidence. In the second part, the Mercury's perihelion advance and other theoretical problems are discussed in relationship with the GR physical foundations. Controversies in literature devoted to the GR tests are analyzed. The unified GR approach to particles and photons is discussed with the emphasis on the GR classical tests. Finally, the alternative theory of relativistic effect treatment is presented.

For the past thirty years, intense efforts have been made to record atomic scale movies that reveal the movement of atoms in molecules, the fast dynamical processes in biological tissues and cells, and the changes in the structure of a solid confined to nano-scale volumes. A combination of sub-nanometer spatial resolution with picosecond or even femtosecond temporal resolution is required for such atomic movies. Additional important information can be obtained when the energy of the electron beam transmitted through the sample is measured. The four dimensional (4D) spatially and temporally resolved ultrafast electron microscopy method is made possible by the extremely high detection efficiency that is reached in 4D electron microscopy. Using ultra-short electron bunches for the visualization of biological tissue can also improve the spatial resolution compared to conventional electron microscopes, thereby enabling the study of complex biological samples of relevance to the life sciences. Of particular interest to a broad audience is the possibility to create a video, and in the future, a real atomic movie, using 4D electron tomography.

Abstract:
The dielectric spectrum of liquid water, $10^{4} - 10^{11}$ Hz, is interpreted in terms of diffusion of charges, formed as a result of self-ionization of H$_{2}$O molecules. This approach explains the Debye relaxation and the dc conductivity as two manifestations of this diffusion. The Debye relaxation is due to the charge diffusion with a fast recombination rate, $1/\tau_{2}$, while the dc conductivity is a manifestation of the diffusion with a much slower recombination rate, $1/\tau_{1}$. Applying a simple model based on Brownian-like diffusion, we find $\tau_{2} \simeq 10^{-11}$ s and $\tau_{1} \simeq 10^{-6}$ s, and the concentrations of the charge carriers, involved in each of the two processes, $N_{2} \simeq 5 \times 10^{26}$ m$^{-3}$ and $N_{1} \simeq 10^{14}$ m$^{-3}$. Further, we relate $N_{2}$ and $N_{1}$ to the total concentration of H$_{3}$O$^{+}$--OH$^{-}$ pairs and to the pH index, respectively, and find the lifetime of a single water molecule, $\tau_{0} \simeq 10^{-9}$ s. Finally, we show that the high permittivity of water results mostly from flickering of separated charges, rather than from reorientations of intact molecular dipoles.

For any finite-dimensional complex semisimple Lie algebra, two ellipsoids (primary and secondary) are considered. The equations of these ellipsoids are Diophantine equations, and the Weyl group acts on the sets of all their Diophantine solutions. This provides two realizations (primary and secondary) of the Weyl group on the sets of Diophantine solutions of the equations of the ellipsoids. The primary realization of the Weyl group suggests an order on the Weyl group, which is stronger than the Chevalley-Bruhat ordering of the Weyl group, and which provides an algorithm for the Chevalley-Bruhat ordering. The secondary realization of the Weyl group provides an algorithm for constructing all reduced expressions for any of its elements, and thus provides another way for the Chevalley-Bruhat ordering of the Weyl group.

Abstract:
The ionization constant of water Kw is currently determined on the proton conductivity sigma1 which is measured at frequencies lower than 10^7 Hz. Here, we develop the idea that the high frequency conductivity sigma2 (~10^11 Hz), rather than sigma1 represents a net proton dynamics in water, to evaluate the actual concentration c of H3O+ and OH- ions from sigma2. We find c to be not dependent on temperature to conclude that i) water electrodynamics is due to a proton exchange between H3O+ (or OH-) ions and neutral H2O molecules rather than spontaneous ionization of H2O molecules, ii) the common Kw (or pH) reflects the thermoactivation of the H3O+ and OH- ions from the potential of their interaction, iii) the lifetime of a target water molecule does not exceed parts of nanosecond.

Abstract:
A nonlinear differential equation with delay serving as a mathematical model of several applied problems is considered. Sufficient conditions for the global asymptotic stability and for the existence of periodic solutions are given. Two particular applications are treated in detail. The first one is a blood cell production model by Mackey, for which new periodicity criteria are derived. The second application is a modified economic model with delay due to Ramsey. An optimization problem for a maximal consumption is stated and solved for the latter.