A non-homogeneous boundary-value problem for the Korteweg-de Vries equation in a quarter plane

The Korteweg-de Vries equation was first derived by Boussinesq and Korteweg and de Vries as a model for long-crested small-amplitude long waves propagating on the surface of water. The same partial differential equation has since arisen as a model for unidirectional propagation of waves in a variety of physical systems. In mathematical studies, consideration has been given principally to pure initial-value problems where the wave profile is imagined to be determined everywhere at a given instant of time and the corresponding solution models the further wave motion. The practical, quantitative use of the Korteweg-de Vries equation and its relatives does not always involve the pure initial-value problem. Instead, initial-boundary-value problems often come to the fore. A natural example arises when modeling the effect in a channel of a wave maker mounted at one end, or in modeling near-shore zone motions generated by waves propagating from deep water. Indeed, the initial-boundary-value problem

studied here arises naturally as a model whenever waves determined at an entry point propagate into a patch of a medium for which disturbances are governed approximately by the Korteweg-de Vries equation. The present essay improves upon earlier work on (0.1) by making use of modern methods for the study of nonlinear dispersive wave equations. Speaking technically, local well-posedness is obtained for initial data in the class for \frac34$"> and boundary data in , whereas global well-posedness is shown to hold for when , and for when . In addition, it is shown that the correspondence that associates to initial data and boundary data the unique solution of (0.1) is analytic. This implies, for example, that solutions may be approximated arbitrarily well by solving a finite number of linear problems.

     

Supramolecular stabilization of N(2)H(7)(+)

The cation N(2)H(7)(+) has been stabilized in a largely hydrophobic supramolecular environment and characterized in the solid state. The cation is situated in the bowl-shaped cavity of calix[4]arene. All of the hydrogen atoms are clearly discernible owing to high-quality X-ray data as well as lack of disorder and symmetry-imposed ambiguity. It appears that electrostatic interactions play a critical role in stabilizing the structure.     

Discovery of structurally diverse natural product antagonists of chemokine receptor CXCR3

The chemokines (CXCL9, CXCL10 and CXCL11) and associated CXCR3 receptor are expressed during the inflammatory process from multiple sclerosis, atherosclerosis or organ transplantation resulting in the recruitment of lymphocytes leading to tissue damage. It is hypothesized that blocking of the ligand/CXCR3 receptor interaction has potential to provide opportunity for development of agents that would block tissue rejection. In this paper, four classes of natural product inhibitors (IC₅₀ ranging 0.1–41 M) have been described that block the CXCR3 receptor interaction of IP-10 ligand. These include a cyclic thiopeptide (duramycin), polyketide glycosides (roselipins), steroidal glycosides (hypoglausin A and dioscin) and a novel alkyl pyridinium alkaloid that were isolated by bioassay-guided fractionation of the organic extracts derived from actinomycete, fungal, plant and marine sources and discovered using ¹²⁵ I IP-10/CXCR3 binding assay. Duramycin was the most potent with an IC₅₀ of 0.1 M. Roselipins 2A, 2B and 1A showed IC₅₀ values of 14.6, 23.5, and 41 M, respectively. Diosgenin glycosides dioscin, hypoglaucin A and kallstroemin D exhibited IC₅₀ values of 2.1, 0.47 and 3 M, respectively. A novel cyclic 3-alkyl pyridinium salt isolated from a sponge displayed a binding IC₅₀ of 0.67 M.     

Oxabicyclo[3.2.1]octane derivatives as highly reactive dienophiles: synthesis of bicyclo[5.n.0] systems

[reaction: see text] We have developed highly versatile, homochiral oxabicyclo[3.2.1]octadiene building blocks for the synthesis of natural products. We have found that these bridged alkenes undergo exceptionally facile Diels-Alder reactions and react faster than several well studied bicyclo[2.2.1]heptene dienophiles. The reaction proceeds with high levels of stereochemical control and in very good to excellent yields, providing access to bicyclo[5.4.0]undecane and bicyclo[5.3.0]decane systems. This reactivity is attributed to strain and homoconjugation effects.