Crystal and molecular structure of 5,5-dimethyl-δ1-1,2,4-triazolin-3-one

The spectroscopic and X-ray data of the title compound are presented and discussed. Some anomalies are attributed to the lack of conjugation between the diazene and carbonyl group, and to the association of the molecules in the solid state through an unusual intermolecular hydrogen bond between the amide nitrogen atom and the diazene nitrogen atom N2.     

On some nonself mappings

Let X be a Banach space, let K be a non–empty closed subset of X and let T : K → X be a non–self mapping. The main result of this paper is that if T satisfies the contractive–type condition (1.1) below and maps ?K (?K the boundary of K) into K then T has a unique fixed point in K.     

Global estimates and energy identities for elliptic systems with antisymmetric potentials

We derive global estimates in critical scale invariant norms for solutions of elliptic systems with antisymmetric potentials and almost holomorphic Hopf differential in two dimensions. Moreover, we obtain new energy identities in such norms for sequences of solutions of these systems. The results apply to harmonic maps into general target manifolds and surfaces with prescribed mean curvature. In particular, the results confirm a conjecture of Rivière in the two-dimensional setting.     

Piaselenol—Piaselenolium—Pentaiodid (C6H4N2Se · C6H5N2Se+I3− · I2), eine Struktur mit Polyiodid-schichten

Piaselenole—Piaselenolium—Pentaiodide (C₆H₄N₂Se · C₆H₅N₂Se⁺ I₃^? · I₂), a Structure with Polyiodide Layers The title compound crystallizes in the monoclinic space group P2₁/n with a = 9.320(3), b = 13.812(2), c = 17.159(3) Å, β = 96.11(2)°, V = 2196.3 ų, Z = 4. There occur no isolated I^5? anions but layer-shaped polyiodide aggregates built up by linear, asymmetric I₃^? anions and I₂ molecules. Almost linear triiodide chains are connected by I₂ molecules in a novel type of arrangement to form slightly puckered layers. The polyiodide layers contain several substructures known from other examples. The piaselenole and its conjugated acid, the piaselenolium cation, form a ribbon-like quasi-polymer in which the two components are alternating. They are connected in turns by a linear NH? N hydrogen bridge (N? N: 2.844 Å) and by a so called (SeN)₂-connectivity parallelogram, in which Se? N bonds and Se? N contacts are adjacent. Here we found a very short Se? N contact distance of 2.691 Å. The bond distances of piaselenole (Se? N: 1.787(3) Å, N? C: 1.318(5) Å, C? C: 1.453(8) Å) and also the angles are equal or similar to those occuring in other 1,2,5-selenadiazoles. The protonation of one N in the SeN₂ unit results in a loss of symmetry and significant changes in bonding distances and angles.     

Localized partial traps in diffusion processes and random walks

Reaction-diffusion equations, in which the reaction is described by a sink term consisting of a sum of delta functions, are studied. It is shown that the Laplace transform of the reactive Green's function can be analytically expressed in terms of the Green's function for diffusion in the absence of reaction. Moreover, a simple relation between the Green's functions satisfying the radiation boundary condition and the reflecting boundary condition is obtained. Several applications are presented and the formalism is used to establish the relationship between the time-dependent geminate recombination yield and the bimolecular reaction rate for diffusion-influenced reactions. Finally, an analogous development for lattice random walks is presented.