Stark's conjecture over complex cubic number fields

Systematic computation of Stark units over nontotally real base fields is carried out for the first time. Since the information provided by Stark's conjecture is significantly less in this situation than the information provided over totally real base fields, new techniques are required. Precomputing Stark units in relative quadratic extensions (where the conjecture is already known to hold) and coupling this information with the Fincke-Pohst algorithm applied to certain quadratic forms leads to a significant reduction in search time for finding Stark units in larger extensions (where the conjecture is still unproven). Stark's conjecture is verified in each case for these Stark units in larger extensions and explicit generating polynomials for abelian extensions over complex cubic base fields, including Hilbert class fields, are obtained from the minimal polynomials of these new Stark units.

     

General covariance of the non-abelian DBI-action: checks and balances

We perform three tests on our proposal to implement diffeomorphism invariance in the non-abelian D0-brane DBI action as a base-point independence constraint between matrix Riemann normal coordinate systems. First, we show that T-duality along an isometry correctly interchanges the potential and kinetic terms in the action. Second, we show that the method to impose base-point independence using an auxiliary dN²-dimensional nonlinear sigma model also works for metrics which are curved along the brane, provided a physical gauge choice is made at the end. Third, we show that without alteration this method is applicable to higher order in velocities. Testing specifically to order 4, we elucidate the range of validity of the symmetrized trace approximation to the non-abelian DBI action.     

Solutions of the WDVV Equations and Integrable Hierarchies of KP Type

We show that reductions of KP hierarchies related to the loop algebra of SL_n with homogeneous gradation give solutions of the Darboux-Egoroff system of PDE's. Using explicit dressing matrices of the Riemann-Hilbert problem generalized to include a set of commuting additional symmetries, we construct solutions of the Witten– Dijkgraaf–E. Verlinde–H. Verlinde equations.     

Rank,covering number and a simple spectrum

For a dynamical system (X, B, T, μ) we investigate the connections between metric invariants, the rankr(T) and the covering numberF ^*(T) and a spectral property for having a simple spectrum. Given a positive integerr≥2, a real numberb, 0<b<1 such thatr·b≥1, we construct examples of systems withr(T)=r, F ^*(T)=b and having a simple spectrum.     

(Systemic) phototoxicity of drugs and other xenobiotics.

Xenobiotics extensively used in drugs, cosmetics, food and agricultural chemicals can produce adverse biological effects. These toxic effects are separated into classes, e.g. hepatotoxicity, genotoxicity and neurotoxicity. Skin allergy, part of immunotoxicity, is also a subdivision of toxicology. When light is an essential condition for toxicity, the xenobiotic is called phototoxic. Thus it fits into the logic of toxicology that photoallergic compounds are a subdivision of phototoxic compounds. Phototoxicons as a group do not differ from the group of phototherapeutics with regard to their eventual biological effects. The primary photoreactions, secondary molecular processes, biomolecules involved and cellular and tissue damage are similar. The difference between the two groups is in the appreciation of the photobiological effects: adverse vs. desired. The aim of research is to determine the part of the molecular structure which makes a given compound phototoxic. With that knowledge the structure of the phototoxicon can be changed. This can result in a derivative which still has the desired properties of the parent compound, but is no longer phototoxic. This aim can be reached by combining data from both in vitro and in vivo research. The variety and number of phototoxic compounds is large. This, together with the limited research effort devoted to this subject so far, means that for most phototoxic xenobiotics a relationship between structure and in vivo photoreactivity is not available. In this review, emphasis is placed on xenobiotics whose in vitro and in vivo photochemistry have been studied. Furthermore, possible phototoxic effects which do not concern the skin but involve inner organs (systemic effects) are considered. References in this review mostly concern investigations over the last 10 years. For older literature or for additional information, references to other reviews are given. Important groups of phototoxic xenobiotics not dealt with in this article were already sufficiently covered in the reviews referred to.