We will review the use of noncommutative topology in the generalisation of Bloch theory from crystals to quasicrystals. After introducing Bloch theory, we will construct the noncommutative space of tilings and we will argue that this is the noncommutative analogue of the Brillouin zone which is used in Bloch theory. The K-theory of the noncommutative Brillouin zone will be used to provide a labeling of the gaps in the spectrum of quasiperiodic Hamiltonians, which can be seen as first step towards a generalisation of Bloch theory to quasicrystals.

Skew-self-adjoint discrete and continuous Dirac type systems: inverse problems and Borg-Marchenko theorems

Topics in the theory of positive solutions of second-order elliptic and parabolic partial differential equations

Topics in the theory of positive solutions of second-order elliptic and parabolic partial differential equations

It would indeed be remarkable if Nature fortified herself against further advances in knowledge behind the analytical difficulties of the many-body problem. — Max Born, 1960 We present here an introduction to Hartree-Fock theory in Quantum Chemistry. From the molecular Hamiltonian, using and discussing the Born-Oppenheimer approximation, we arrive to the Hartree and Hartree-Fock equations for the electronic problem. Special emphasis is placed in the most relevant mathematical aspects of the theoretical derivation of the final equations, as well as in the results regarding the existence and uniqueness of their solutions. Additionally, the discretization of the one-electron orbitals space is reviewed and the Roothaan-Hall formalism introduced. Finally, the basic underlying concepts related to the construction and selection of Gaussian basis sets are discussed focusing in algorithmic efficiency issues. The whole work is intentionally introductory and rather self-contained, so that it may be useful for non experts that aim to use quantum chemical methods in interdisciplinary applications. Moreover, much material that is found scattered in the literature has been put together here to facilitate comprehension and to serve as a handy reference.

Global stability of travelling fronts for a damped wave equation with bistable nonlinearity

Weyl functions, inverse problem and special solutions for the system auxiliary to the nonlinear optics equation