Abstract:
We introduce a concept of a fractional-derivatives series and prove that any linear partial differential equation in two independent variables has a fractional-derivatives series solution with coefficients from a differentially closed field of zero characteristic. The obtained results are extended from a single equation to $D$-modules having infinite-dimensional space of solutions (i. e. non-holonomic $D$-modules). As applications we design algorithms for treating first-order factors of a linear partial differential operator, in particular for finding all (right or left) first-order factors.

Abstract:
Public-key cryptosystems are suggested based on invariants of groups. We give also an overview of the known cryptosystems which involve groups.

Abstract:
We investigate the monotonic growth of longitudinal interlayer magnetoresistance $\bar{R}_{zz}(B_z) $, analytically and numerically in the self-consistent Born approximation. We show that in a weak magnetic field the monotonic part of $\bar{R}_{zz}(B_z)$ is almost constant and starts to grow only above the crossover field $B_{c}$, when the Landau levels (LL) become isolated, i.e. when the LL separation becomes greater than the LL broadening. In higher field $B_{z}>>B_{c}$, $\bar{R}_{zz}(B_{z}) \propto B_{z}^{1/2}$ in agreement with previous works.

Abstract:
The analytical and numerical study of the angular dependence of magnetoresistance in layered quasi-two-dimensional (Q2D) metals is performed. The harmonic expansion analytical formulas for the angular dependence of Fermi-surface cross-section area in external magnetic field are obtained for various typical crystal symmetries. The simple azimuth-angle dependence of the Yamaji angles is derived for the elliptic in-plane Fermi surface. These formulas correct some previous results and allow the simple and effective interpretation of the magnetic quantum oscillations data in cuprate high-temperature superconducting materials, in organic metals and other Q2D metals. The relation between the angular dependence of magnetoresistance and of Fermi-surface cross-section area is derived. The applicability region of all results obtained and of some previous widely used analytical results is investigated using the numerical calculations.

Abstract:
We investigate the properties and the microscopic structure of superconductivity (SC), coexisting and sharing the common conducting band with density wave (DW). Such coexistence may take place when the nesting of the Fermi surface (FS) is not perfect, and in the DW state some quasi-particle states remain on the Fermi level and lead to the Cooper instability. The dispersion of such quasi-particle states is, in general, very different from that without DW. Therefore, the properties of SC on the DW background may strongly differ from those without DW. The upper critical field $H_{c2}$ in such a SC state increases as the system approaches the critical pressure, where the ungapped quasi-particles and superconductivity just appear, and it may considerably exceed the usual $H_{c2}$ value without DW. The SDW background strongly suppresses the singlet SC pairing, while it does not affect so much the triplet SC transition temperature. The results obtained explain the experimental observations in layered organic metals (TMTSF)$_{2}$PF$_{6}$ and $\alpha $-(BEDT-TTF)$_{2}$KHg(SCN)$_{4}$, where SC appears in the DW states under pressure and shows many unusual properties.

Abstract:
Superconductivity (SC) may microscopically coexist with density wave (DW) when the nesting of the Fermi surface (FS) is not perfect. There are, at least, two possible microscopic structures of a DW state with quasi-particle states remaining on the Fermi level and leading to the Cooper instability: (i) the soliton-wall phase and (ii) the small ungapped Fermi-surface pockets. The dispersion of such quasi-particle states strongly differs from that without DW, and so do the properties of SC on the DW background. The upper critical field $H_{c2}$ in such a SC state strongly increases as the system approaches the critical pressure, where superconductivity first appears. $H_{c2}$ may considerably exceed its typical value without DW and has unusual upward curvature as function of temperature. The results obtained explain the experimental observations in layered organic superconductors (TMTSF)$_{2}$PF$_{6}$ and $\alpha $-(BEDT TTF)$_{2}$KHg(SCN)$_{4}$.

Abstract:
It is shown that in rather strong magnetic field the interlayer electron conductivity is exponentially damped by the Coulomb barrier arising from the formation of polaron around each localized electron state. The theoretical model is developed to describe this effect, and the calculation of the temperature and field dependence of interlayer magnetoresistance is performed. The results obtained agree well with the experimental data in GaAs/AlGaAs heterostructures and in strongly anisotropic organic metals. The proposed theory allows to use the experiments on interlayer magnetoresistance to investigate the electron states, localized by magnetic field and disorder.

Abstract:
The Shubnikov - de Haas effect in quasi-two-dimensional normal metals is studied. The interlayer conductivity is calculated using the Kubo formula. The electron scattering on short-range is considered in the self-consistent Born approximation. The result obtained differs from that derived from the Boltzmann transport equation. This difference is shown to be a general feature of conductivity in magnetic field. A detailed description of the two new qualitative effects -- the field-dependent phase shift of beats and of the slow oscillations of conductivity is provided. The results obtained are applicable to strongly anisotropic organic metals and to other quasi-two-dimensional compounds.

Abstract:
The longitudinal interlayer magnetoresistance $R_{zz}(B_{z})$ is calculated in strongly anisotropic layered metals, when the interlayer band width $4t_{z}$ is less than the Landau level separation $\hbar \omega_{c}$. The impurity scattering has much stronger effect in this regime than in 3D metals and leads to a linear longitudinal interlayer magnetoresistance $R_{zz}\propto B_{z}$ in the interval $\hbar \omega_{c}>4t_{z}>>\sqrt{\Gamma_{0}\hbar \omega_{c}}$ changing to a square-root dependence $R_{zz}\propto B_{z}^{1/2}$ at higher field or smaller $t_{z}$. The crossover field allows to estimate the interlayer transfer integral as $t_{z}\sim \sqrt{\Gamma_{0}\hbar \omega_{c}}$. Longitudinal interlayer magnetoresistance, being robust to the increase of temperature or long-range disorder, is easy for measurements and provides a useful tool to investigate the electronic structure of quasi-two-dimensional compounds.

Abstract:
We investigate the conductivity in layered metals in magnetic field in the weakly incoherent limit, when the interlayer transfer integral is smaller than the Landau level broadening due to the impurity potential, but the interlayer electron tunnelling conserves the intralayer momentum. It is shown that the impurity potential has much stronger effect in this regime, than in the quasi-2D metals in the coherent limit. The weakly incoherent regime has several new qualitative features, not found in the previous theoretical approaches. The background interlayer magnetoresistance in this regime monotonically grows with increasing of magnetic field perpendicular to the conducting layers. The effective electron mean free time is considerably shorter than in the coherent regime and decreases with magnetic field. This enhances the role of higher harmonics in the angular magnetoresistance oscillations and increases the Dingle temperature, which damps the magnetic quantum oscillations.