Timelike Minimal Surfaces via Loop Groups

This work consists of two parts. In Part I, we shall give a systematic study of Lorentz conformal structure from structural viewpoints. We study manifolds with split-complex structure. We apply general results on split-complex structure for the study of Lorentz surfaces.In Part II, we study the conformal realization of Lorentz surfaces in the Minkowski 3-space via conformal minimal immersions. We apply loop group theoretic Weierstrass-type representation of timelike constant mean curvature for timelike minimal surfaces. Classical integral representation formula for timelike minimal surfaces will be recovered from loop group theoretic viewpoint.     

Gauss Algebras

A special class of associative algebras is introduced and studied. A criteria of semisimplicity for suitable category of modules is obtained.     

Some Problems of Harmonic Analysis on Reductive Groups

An algebraic approach to harmonic analysis on reductive Lie groups is proposed. The case of complex semisimple Lie groups is considered in details. Some problems for real reductive Lie groups are discussed.     

On the Integrability Conditions for Some Structures Related to Evolution Differential Equations

Using the result by D. Gessler, we show that any invariant variational bivector (resp., variational 2-form) on an evolution equation with nondegenerate right-hand side is Hamiltonian (resp., symplectic).     

Explicit 2-Factorisations of the Odd Graph

In this note we show how 1-factors in the middle two layers of the discrete cube can be used to construct 2-factors in the Odd graph (the Kneser graph of (k − 1)-sets from a (2k − 1)-set). In particular, we use the lexical matchings of Kierstead and Trotter, and the modular matchings of Duffus, Kierstead and Snevily, to give explicit constructions of two different 2-factorisations of the Odd graph. This revised version was published online in September 2006 with corrections to the Cover Date.     

The long‐time behaviour of the thermoconvective flow in a porous medium

For the Boussinesq approximation of the equations of coupled heat and fluid flow in a porous medium we show that the corresponding system of partial differential equations possesses a global attractor. We give lower and upper bounds of the Hausdorff dimension of the attractor depending on a physical parameter of the system, namely the Rayleigh number of the flow. Numerical experiments confirm the theoretical findings and raise new questions on the structure of the solutions of the system. Copyright © 2004 John Wiley & Sons, Ltd.     

Some non‐standard problems in viscous flow

Energy bounds are derived for Dirichlet type boundary value problems for the Navier–Stokes and Stokes equations when a combination of the solution values initially and at a later time is prescribed. The bounds are obtained by means of a differential inequality and imply uniqueness and continuous data dependence of the solutions for a range of values of the parameter in the non‐standard auxiliary condition. Copyright © 2004 John Wiley & Sons, Ltd.     

Anionically Substituted 1,1′,1″‐Methylidynetris[1H‐pyrazole] Ligands for the Formation of Neutral Lanthanide Complexes in Water: Synthesis,Characterization, and Photophysical Properties

The six‐step synthesis of the new podand‐type ligand 6,6′,6″‐[methylidenetri(1H‐pyrazole‐1,3‐diyl)]tris[pyridine‐2‐carboxylic acid] (LH₃) is described. Reaction of LH₃ with LnCl₃ ?6 H₂O (Ln=Eu, Gd, Tb) in MeOH resulted in the isolation of [LnL]?HCl complexes characterized by elemental analysis, mass and IR spectroscopy. Photophysical studies of the Eu and Tb complexes in aqueous solutions revealed the characteristic luminescence features of the metal atoms, indicative of an efficient ligand‐to‐metal energy‐transfer process. Determination of the luminescence quantum yields in H₂O showed the Tb complex to be highly luminescent (?=15%), while, for the Eu complex, the quantum efficiency was only 2%. Excited‐state‐lifetime measurements in H₂O and D₂O evidenced the presence of ca. three H₂O molecules in the first coordination sphere of the complexes. Investigation of the Gd complex allowed the determination of the ligand‐centered triplet state and showed the ligand to be well suited for energy transfer to the metal. The luminescence properties of the complexes are described, and the properties of the ligand as a suitable complexation pocket is questioned.     

Non-protein amino acids in peptide design

An overview of the use of non-protein amino acids in the design of conformationally well-defined peptides, based on work from the author’s laboratory, is discussed. The crystal structures of several designed oligopeptides illustrate the useα-aminoisobutyric acid (Aib) in the construction of helices, D-amino acids in the design of helix termination segments and^DPro-Xxx segments for nucleating ofβ-hairpin structures.β- andγ-amino acid residues have been used to expand the range of designed polypeptide structures. Dedicated to Professor C N R Rao on his 70th birthday     

A new reaction of vicinal sulfonyliminocarboxylates with phosphites

The reaction of alkyl trifluoro(organylsulfonylimino)propionates with phosphites occurs with NC transfer of the RSO₂ group and leads to sulfonyl-substituted trifluoroalanine derivatives. The novel rearrangement is interpreted as cheletropic 1,4-cycloaddition of the phosphite and subsequent 1,2-shift of the sulfonyl group in the intermediate cyclic phosphorane.