The exponential localization of Wannier functions in two or three dimensions is proven for all insulators that display time-reversal symmetry, settling a long-standing conjecture. Our proof relies on the equivalence between the existence of analytic quasi-Bloch functions and the nullity of the Chern numbers (or of the Hall current) for the system under consideration. The same equivalence implies that Chern insulators cannot display exponentially localized Wannier functions. An explicit condition for the reality of the Wannier functions is identified.

A detailed account of the dependence of x-ray absorption spectra (XAS) on polarisation and light beam directions is given. Anisotropic XAS measured on single crystals, layered compounds, polymers and oriented powders are tabulated. The x-ray absorption cross section by non-magnetic samples is derived, including electric dipole and quadrupole contributions. Orders of magnitude are found for magnetic dipole and electric dipoleoctupole terms. Using a spherical tensor expansion of the absorption cross section, simple analytical formulae are given for the angular dependence of the electric dipole and quadrupole contributions to XAS for all crystal point groups. These formulae are valid down to the edge and include many-body effects. They are applied to two experimental examples, the iron K-edge in haematite and the copper K-edge in CuCI:-. For the latter case, spherical tensor components are interpreted in terms of molecular orbitals. The influence of unpolarised and circularly polarised x-rays is discussed, as well as various experimental problems met in angle-resolved XAS. In an appendix, the quadrupole contribution to angular XAS is derived within the multiple-scattering formalism, along with closedexpressions for the dipole and quadrupole spherical tensor components M O LlI,lII-edge Nb K-edge Nd L,,,-edge YB a,Cu,O, KBr KBr0, Graphite Cd NaCl KCI KCIO; CO"'(NH,)~(C~OJ)~C~ . KCI Cr'v(5,10,15,20-tetra-p-tolylporphyrin)0 CrV(5,10,15,20-tetraptoly1porphyrin)N c u (~reatinium)~CuCI, CuCI,. 2 H 2 0 Cu( l-methyIimidaz~le)~Cl~ Cu( 1 ,2-dimethylimidazole),C12 Plastocyanin C~(2-methyIpyridine)~Cl~ Cu(imidazole)2C12 Cu(imidazole), Cu(imidazole), . 2N03 Cu( 1 ,3,5-trimethylpyrazole),BF4

The subject of this paper are two Hopf algebras which are the non-commutative analogues of two different groups of formal power series. The first group is the set of invertible series with the group law being multiplication of series, while the second group is the set of formal diffeomorphisms with the group law being composition of series. The motivation to introduce these Hopf algebras comes from the study of formal series with non-commutative coefficients. Invertible series with non-commutative coefficients still form a group, and we interpret the corresponding new non-commutative Hopf algebra as an alternative to the natural Hopf algebra given by the co-ordinate ring of the group, which has the advantage of being functorial in the algebra of coefficients. For the formal diffeomorphisms with non-commutative coefficients, this interpretation fails, because in this case the composition is not associative anymore. However, we show that for the dual non-commutative algebra there exists a natural co-associative co-product defining a non-commutative Hopf algebra. Moreover, we give an explicit formula for the antipode, which represents a non-commutative version of the Lagrange inversion formula, and we show that its coefficients are related to planar binary trees. Then we extend these results to the semi-direct co-product of the previous Hopf algebras, and to series in several variables. Finally, we show how the non-commutative Hopf algebras of formal series are related to some renormalization Hopf algebras, which are combinatorial Hopf algebras motivated by the renormalization procedure in quantum field theory, and to the renormalization functor given by the double tensor algebra on a bi-algebra.