Artificial time integration

Many recent algorithmic approaches involve the construction of a differential equation model for computational purposes, typically by introducing an artificial time variable. The actual computational model involves a discretization of the now time-dependent differential system, usually employing forward Euler. The resulting dynamics of such an algorithm is then a discrete dynamics, and it is expected to be “close enough” to the dynamics of the continuous system (which is typically easier to analyze) provided that small – hence many – time steps, or iterations, are taken. Indeed, recent papers in inverse problems and image processing routinely report results requiring thousands of iterations to converge. This makes one wonder if and how the computational modeling process can be improved to better reflect the actual properties sought. In this article we elaborate on several problem instances that illustrate the above observations. Algorithms may often lend themselves to a dual interpretation, in terms of a simply discretized differential equation with artificial time and in terms of a simple optimization algorithm; such a dual interpretation can be advantageous. We show how a broader computational modeling approach may possibly lead to algorithms with improved efficiency. AMS subject classification (2000)  65L05, 65M32, 65N21, 65N22, 65D18     

Writhing instabilities of twisted rods: from infinite to finite length

We use three different approaches to describe the static spatial configurations of a twisted rod as well as its stability during rigid loading experiments. The first approach considers the rod as infinite in length and predicts an instability causing a jump to self-contact at a certain point of the experiment. Semi-finite corrections, taken into account as a second approach, reveal some possible experiments in which the configuration of a very long rod will be stable through out. Finally, in a third approach, we consider a rod of real finite length and we show that another type of instability may occur, leading to possible hysteresis behavior. As we go from infinite to finite length, we compare the different information given by the three approaches on the possible equilibrium configurations of the rod and their stability. These finite size effects studied here in a 1D elasticity problem could help us guess what are the stability features of other more complicated (2D elastic shells for example) problems for which only the infinite length approach is understood.     

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.

     

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.     

(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.