Abstract:
We present results of the digital spectral analysis of the time series containing the daily samples of the atmosphere surface pressure measured at Saratov city from Jan 1, 1995 to Dec 31, 1999. We calculated also cross correlation functions between the pressure fluctuations and the planetary geomagnetic indices, Ap and Cp, and cross correlation function between these fluctuations and the cosmic-ray intensity. Two peaks detected in the calculated power spectrum are related, probably, with the atmosphere tidal waves. The relation of the pressure fluctuations with the Ap index of the geomagnetic activity is statistically insignificant. The cross correlation function between the pressure fluctuations and the Cp index has the maximal value 0.044+-0.023 at the pressure time delay 4 days. The statistically significant negative correlation is discovered between the surface pressure fluctuations and the cosmic-ray intensity. The cross correlation function has two minima: at the zero time delay with the value -0.068+-0.023, and at the pressure time delay 12 days with the value -0.087+-0.023. The negative correlation between the pressure and the cosmic-ray intensity is observed at the pressure time delay as long as 17 days and can be explained by the so-called condensation mechanism of an interaction of the cosmic rays with the atmosphere.

Abstract:
The problem of restoration of the source brightness distribution from an analysis of the stellar and AGNs microlensing light curves is investigated. In case of microlensing of stars by a point-mass lens as well as for caustic crossing events for binary lens the problem can be reduced to solution of the Fredholm integral equation of the 1st kind. Concrete form of the kernel of this equation depends on a type of the microlensing event. Assuming the circular symmetry of the stellar disk the search for radial brightness distribution can be carried out in the special compact sets of functions which correspond to the physics of the problem. These sets include the non-negative functions that are not increasing with increasing distance from the center of stellar disk and the upwards convex non-negative functions. The brightness distribution for the AGNs accretion disks is also circularly symmetric, but only in the locally co-moving frame. Therefore, the kernel of integral equation that determined the AGN microlensing light curve must take into account equally with the projection effect on picture plane the influence of relativistic effects. The search for solution of this equation can be carried out in the set of non-negative down convex functions. The results of analysis of microlensing light curves for the red giant MACHO Alert 95-30 and the A6 star MACHO 98-SMC-1 as well as the results of numerical simulations for the AGN microlensing observations are given.

Abstract:
For an analysis of microlensing observational data in case of binary gravitational lenses as well as for an interpretation of observations of high magnification events in multiple images of a lensed quasar it is necessary to calculate for a given source the microlensing light curve by a fold caustic. This problem comes to the numerical calculation of a singular integral. We formulated the sufficient condition of a convergence of the integral sum for this singular integral. The strictly approach to the problem of a comparison of model results with the unequally sampled observational data consists in calculation of the model light curve in equidistant points of the canonical dissection of the integration segment and a following interpolation of its values at the moments of observations.

Abstract:
Therapeutic response to selective serotonin (5-HT) reuptake inhibitors in Major Depressive Disorder (MDD) varies considerably among patients, and the onset of antidepressant therapeutic action is delayed until after 2 to 4 weeks of treatment. The objective of this study was to analyze changes within methoxyindole and kynurenine (KYN) branches of tryptophan pathway to determine whether differential regulation within these branches may contribute to mechanism of variation in response to treatment. Metabolomics approach was used to characterize early biochemical changes in tryptophan pathway and correlated biochemical changes with treatment outcome. Outpatients with MDD were randomly assigned to sertraline (n = 35) or placebo (n = 40) in a double-blind 4-week trial; response to treatment was measured using the 17-item Hamilton Rating Scale for Depression (HAMD17). Targeted electrochemistry based metabolomic platform (LCECA) was used to profile serum samples from MDD patients. The response rate was slightly higher for sertraline than for placebo (21/35 [60%] vs. 20/40 [50%], respectively, χ2(1) = 0.75, p = 0.39). Patients showing a good response to sertraline had higher pretreatment levels of 5-methoxytryptamine (5-MTPM), greater reduction in 5-MTPM levels after treatment, an increase in 5-Methoxytryptophol (5-MTPOL) and Melatonin (MEL) levels, and decreases in the (KYN)/MEL and 3-Hydroxykynurenine (3-OHKY)/MEL ratios post-treatment compared to pretreatment. These changes were not seen in the patients showing poor response to sertraline. In the placebo group, more favorable treatment outcome was associated with increases in 5-MTPOL and MEL levels and significant decreases in the KYN/MEL and 3-OHKY/MEL; changes in 5-MTPM levels were not associated with the 4-week response. These results suggest that recovery from a depressed state due to treatment with drug or with placebo could be associated with preferential utilization of serotonin for production of melatonin and 5-MTPOL.

The influence of fluctuations of atmospheric pressure on the functional state of the humans was studied during spring, autumn and winter seasons. Sensory-motor reaction time and selfreported wellbeing, activity and mood were used for evaluation of functional state. The inter-individual variations of those parameters were compared to meteorological parameters using rank order correlation and general linear model. It was found that atmospheric pressure fluctuations have a stronger negative effect with periods of 120-1200 s and 20-120 s on psychological self-assessment and with periods of 10-20 s and 5-10 s on sensory-motor reaction time than the fluctuations with other periods.

Abstract:
Unifying hierarchies of integrable equations are discussed. They are constructed via generalized Hirota identity. It is shown that the Combescure transformations, known for a long time for the Darboux system and having a simple geometrical meaning, are in fact the symmetry transformations of generalized integrable hierarchies. Generalized equation written in terms of invariants of Combescure transformations are the usual integrable equations and their modified partners. The KP-mKP, DS-mDS hierarchies and Darboux system are considered.

Abstract:
A general scheme is proposed for introduction of lattice and q-difference variables to integrable hierarchies in frame of $\bar{\partial}$-dressing method . Using this scheme, lattice and q-difference Darboux-Zakharov-Manakov systems of equations are derived. Darboux, B\"acklund and Combescure transformations and exact solutions for these systems are studied.

Abstract:
Second heavenly equation hierarchy is considered using the framework of hyper-K\"ahler hierarchy developed by Takasaki. Generating equations for the hierarchy are introduced, they are used to construct generating equations for reduced hierarchies. General $N$-reductions, logarithmic reduction and rational reduction for one of the Lax-Sato functions are discussed. It is demonstrated that rational reduction is equivalent to the symmetry constraint.

Abstract:
Analytic-bilinear approach for construction and study of integrable hierarchies, in particular, the KP hierarchy is discussed. It is based on the generalized Hirota identity. This approach allows to represent generalized hierarchies of integrable equations in a condensed form of finite functional equations. Resolution of these functional equations leads to the $\tau$-function and addition formulae to it. General discrete transformations of the $\tau$-function are presented in the determinant form. Closed one-form and other formulae also arise naturally within the approach proposed. Generalized KP hierarchy written in terms of different invariants of Combescure symmetry transformations coincides with the usual KP hierarchy and the mKP hierarchy.

Abstract:
We introduce a general setting for multidimensional dispersionless integrable hierarchy in terms of differential $m$-form $\Omega_m$ with the coefficients satisfying the Pl\"ucker relations, which is gauge-invariantly closed and its gauge-invariant coordinates (ratios of coefficients) are (locally) holomorphic with respect to one of the variables (the spectral variable). We demonstrate that this form defines a hierarchy of dispersionless integrable equations in terms of commuting vector fields locally holomorphic in the spectral variable. The equations of the hierarchy are given by the gauge-invariant closedness equations.