After presenting an intuitive picture of quasi-particle transport in mesoscopic superconductors, which emphasizes the intimate relation between Andreev scattering and zero resistivity, the authors develop a general theory of DC transport in mesoscopic normal-superconducting structures. Generalized multi-probe conductance formulae are derived, which take into account not only the effect of Andreev scattering on transport coefficients, but also the non-conservation of quasi-particle charge which arises in the presence of a superconducting condensate. Experiments on quasi-particle charge imbalance are described naturally by this approach.

We examine equivalent forms of the current-voltage relationship applicable to mesoscopic superconductors and use one of these to derive general conditions for the sign of the conductance in a four-probe measurement. For disordered systems, with well separated voltage probes, where the n o d-s t a t e conductance is positive, we predict that the sign of the 1ongiNdinal four-probe conductance of superconducting wires can be reversed by varying an applied magnetic field, and that at certain critical values of the field. the conductance passes through a singularity.

The influence of bulk superconducting substrates and coatings on the domain structure parameters of a ferromagnetic ultra-thin film was analysed, assuming the validity of the London equation in the superconducting volumes. The problem was analysed for both easy plane and perpendicular easy axis films; however, as the effects of the superconducting substrate are relatively unimportant for films magnetized in plane, most of the article is devoted to easy axis films. Expressions for the demagnetization energy were found for arbitrary magnetization distribution. It was shown that a superconducting substrate changes the domain structure properties even if the layer is coated with a non-superconducting material. For sufficiently thin films the domain structure can be fully suppressed, which was shown by considering an isolated Bloch wall for larger anisotropy (Q > 1) and the critical domain structure for low anisotropy (Q < 1). The results of numerical calculations for critical ferromagnetic film thickness (as a function of anisotropy and London penetration depth) are presented.

We analyse the pmblem of one-dimensional quantum mechanics on arbitrary graphs as idealized models for quantum systems on spaces with non-hivial topologies. In panic& we argue that such models can be made to ac"modate the physical characteristics of wavefunctions on a nehvork of wires and offer several derivations of a panicular junction condition. Throughout we adopt a continuity condition for the wavefunction at each primitive node in the network. Results are applied to the problem of the energy specmm of a system containing one and infinitely many junctions.

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