Self-similar intermediate asymptotics for a degenerate parabolic filtration-absorption equation

The equation ∂tu = u∂xx2u − (c − 1)(∂xu)2 is known in literature as a qualitative mathematical model of some biological phenomena. Here this equation is derived as a model of the groundwater flow in a water-absorbing fissurized porous rock; therefore, we refer to this equation as a filtration-absorption equation. A family of self-similar solutions to this equation is constructed. Numerical investigation of the evolution of non-self-similar solutions to the Cauchy problems having compactly supported initial conditions is performed. Numerical experiments indicate that the self-similar solutions obtained represent intermediate asymptotics of a wider class of solutions when the influence of details of the initial conditions disappears but the solution is still far from the ultimate state: identical zero. An open problem caused by the nonuniqueness of the solution of the Cauchy problem is discussed.

Self-assembly of a heteroduplex helicate from two different ligand strands and Cu(II) cations.

Cu(II) ions have been reacted with a 1/1 mixture of two linear ligands, one containing three 2,2'- bipyridine groups and the other three 2,2':6',2"-terpyridine groups. Absorption spectroscopy and fast atom bombardment mass spectrometry indicate the formation of a trinuclear complex containing one ligand of each kind. Determination of the crystal structure of this compound has confirmed that it is indeed a linear trinuclear complex in which two different ligands are wrapped in a helical fashion around the pentacoordinated metal ions. The central coordination geometry is trigonal bipyramidal; the two lateral Cu(II) ions are in a square pyramidal environment. Thus, a heteroduplex helicate is formed by the self-assembly of two different ligand strands and three specific metal ions induced by the coordination number and geometry of the latter. The self-assembly process may be considered to result from the reading of the steric and binding information present in the two ligands by Cu(II) ions through a pentacoordination algorithm. The same ligands have been shown earlier to yield homoduplex helicates from ions of tetrahedral and octahedral coordination geometry and strands of bidentate bipyridines and tridentate terpyridines, respectively. These two types of artificial double helical species may be related on one hand to the natural homoduplex nucleic acids and on the other hand to the DNA:RNA heteroduplex.