Fractional master equations containing fractional time derivatives of order 0 < ω ≤ 1 are introduced on the basis of a recent classification of time generators in ergodic theory. It is shown that fractional master equations are contained as a special case within the traditional theory of continuous time random walks. The corresponding waiting time density ψ(t) is obtained exactly as ψ(t) = (t ω−1 /C)E ω,ω (−t ω /C) where E ω,ω (x) is the generalized Mittag-Leffler function. This waiting time distribution is singular both in the long time as well as in the short time limit. Contents 3 Appendix A. Definition of H-functions 4 References 5

We present an overview of the ONETEP program for linear-scaling density functional theory (DFT) calculations with large basis set (planewave) accuracy on parallel computers. The DFT energy is computed from the density matrix, which is constructed from spatially localized orbitals we call Non-orthogonal Generalized Wannier Functions (NGWFs), expressed in terms of periodic sinc (psinc) functions. During the calculation, both the density matrix and the NGWFs are optimized with localization constraints. By taking advantage of localization, ONETEP is able to perform calculations including thousands of atoms with computational effort, which scales linearly with the number or atoms. The code has a large and diverse range of capabilities, explored in this paper, including different boundary conditions, various exchangecorrelation functionals (with and without exact exchange), finite electronic temperature methods for metallic systems, methods for strongly correlated systems, molecular dynamics, vibrational calculations, time-dependent DFT, electronic transport, core loss spectroscopy, implicit solvation, quantum mechanical (QM)/molecular mechanical and QM-in-QM embedding, density of states calculations, distributed multipole analysis, and methods for partitioning charges and interactions between fragments. Calculations with ONETEP provide unique insights into large and complex systems that require an accurate atomic-level description, ranging from biomolecular to chemical, to materials, and to physical problems, as we show with a small selection of illustrative examples. ONETEP has always aimed to be at the cutting edge of method and software developments, and it serves as a platform for developing new methods of electronic structure simulation. We therefore conclude by describing some of the challenges and directions for its future developments and applications.

We propose a microscopical theory of superconductivity in CuO2 layer within the effective two-band Hubbard model in the strong correlation limit. By applying a projection technique for the matrix Green function in terms of the Hubbard operators, the Dyson equation is derived. It is proved that in the mean-field approximation d-wave superconducting pairing mediated by the conventional exchange interaction occurs. Allowing for the self-energy corrections due to kinematic interaction, a spin-fluctuation d-wave pairing is also obtained. Tc dependence on the hole concentration and kdependence of the gap function are derived. The results show that the exchange interaction (which stems from the interband hopping) prevails over the kinematic interaction (which stems from the intraband hopping).