It is easy to sum chain-free self-energy rainbows, to obtain contributions to anomalous dimensions. It is also easy to resum rainbow-free self-energy chains. Taming the combinatoric explosion of all possible nestings and chainings of a primitive self-energy divergence is a much more demanding problem. We solve it in terms of the coproduct , antipode S, and grading operator Y of the Hopf algebra...

Chiral orbifold models are deened as gauge eld theories with a nite gauge group ?. We start with a conformal current algebra A associated with a connected compact Lie group G and a negative deenite integral invariant bilinear form on its Lie algebra. Any nite group ? of inner automorphisms or A (in particular, any nite subgroup of G) gives rise to a gauge theory with a chiral subalgebra A ? A o...

In the early fties L. Schwartz posed the problem to characterize those linear partial differential operators P(D) that admit a (continuous linear) right inverse on the Fr echet space E(() of all innnitely diierentiable functions on an open set in IR n respectively on the space D 0 (() of all distributions on. This problem was solved in 6], 7]. Its solution was extended to diierential complexes ...

Let be an expansive automorphisms of compact connected abelian group X whose dual group ^ X is cyclic w.r.t. (i.e. ^ X is generated by f n : n 2 Zg for some 2 ^ X). Then there exists a canonical group homomorphism from the spacè 1 (Z;Z) of all bounded two-sided sequences of integers onto X such that = , where is the shift on`1 (Z;Z). We prove that there exists a sooc subshift V ` 1 (Z;Z) such t...

2 ABSTRACT In this paper we explicitly prove the invariance of the time-dependent string gravity Lagrangian with up to four derivatives under the global O(d; d) symmetry.

3 We consider commuting squares of nite dimensional von Neumann algebras having the algebra of complex numbers in the lower left corner. Examples include the vertex models, the spin models (in the sense of subfactor theory) and the commuting squares associated to nite dimensional Kac algebras. To any such commuting square we associate a compact Kac algebra and we compute the corresponding subfa...

The known correspondence between the Kronig{Penney model and certain Jacobi matrices is extended to a wide class of Schrr odinger operators on graphs. Examples include rectangular lattices with and without a magnetic eld, or comb{ shaped graphs leading to a Maryland{type model.

A holey Schrr oder design of type h n 1