Some Decidability and Undecidability Results on Green's Relations for Automatic Monoids

We consider the problem of deciding Green's relations for various types of finitely presented automatic monoids. We establish some decidability as well as some undecidability results.     

Diophantine Undecidability of Function Fields of Characteristic Greater than 2, Finitely Generated over Fields Algebraic over a Finite Field

Let F be a function field of characteristic p > 2, finitely generated over a field C algebraic over a finite field C _p and such that it has an extension of degree p. Then Hilbert's Tenth Problem is not decidable over F.     

On nearly parallel G2-structures

A nearly parallel G₂-structure on a seven-dimensional Riemannian manifold is equivalent to a spin structure with a Killing spinor. We prove general results about the automorphism group of such structures and we construct new examples. We classify all nearly parallel G₂-manifolds with large symmetry group and in particular all homogeneous nearly parallel G₂-structures.     

Hydrogen-bonded block mesogens derived from semiperfluorinated benzoic acids and the non-mesogenic 1,2-bis(4-pyridyl)ethylene

Thermal properties of benzoic acids carrying one or two semiperfluorinated alkoxy tails on the aromatic core have been investigated in binary mixtures with the non-liquid crystalline bidirectional trans-1,2-bis(4-pyridyl)ethylene. The hydrogen bonded complexes built from the complementary molecular species show a significantly enhanced mesophase stability compared with the fluorinated acids in their pure states. The mesophase morphologies of the complexes are governed mainly by the number of the partially fluorinated chains grafted to the acid component. Mixed systems comprising the one-chain acids exhibit a smectic C phase followed by a smectic A phase at more elevated temperatures. Incorporation of a second semiperfluorinated chain into the acid leads to the formation of columnar mesophases. These columnar phases of the H-bonded complexes should represent ribbon phases resulting from the collapse of the smectic layers.     

The ternary system CsCl−NaCl−LaCl3

The ternary system CsCl?NaCl?LaCl₃ was investigated by means of differential thermal analysis and X-ray powder diffraction analysis. There exists one congruently melting compound, Cs₂NaLaCl₆, crystallizing with the cubic elpasolite structure. No quasi-binary section exists for the whole system, however three binaries range from the ternary compound Cs₂NaLaCl₆ to NaCl, CsLa₂Cl₇ and Cs₃LaCl₆ resp., dividing the system in three areas of composition: one triangle, Cs₃LaCl₆?Cs₂NaLaCl₆?CsLa₂Cl₇, containing additionally a compound Cs₂LaCl₅ below 510°C, and the two areas CsCl?NaCl?Cs₂NaLaCl₆?Cs₃LaCl₆ and Cs₂NaLaCl₆?NaCl?LaCl₃?CsLa₂Cl₇, containing a mixed crystal range between LaCl₃ and Na₃La₅Cl₁₈. These areas could be further divided in five triangles, so that the whole system contains six Alkemade triangles.     

Limited stability of the pH reference material sodium tetraborate decahydrate (“borax”)

Thermogravimetric and X-ray powder diffraction analyses revealed that the pH reference material sodium tetraborate decahydrate (“borax”) is not a stable substance but, loosing some of its water of crystallization, transforms slowly into the pentahydrate. The connected pH changes of the solutions are opposite to and larger than, those expected according to the dilution value of the decahydrate as shown by differential potentiometric measurements. The transformation rate depends on the storing time in the closed original container, the time of and after the first exposure of the material to the atmosphere, the frequency of and the relative humidity during subsequent exposures, and the temperature. The transformation reaction is not understood in detail. However, the experiments showed that sodium tetraborate decahydrate is a reliable pH reference material, whose pH is constant within ΔpH=±0.005 within at least two years after preparation, even if occasionally exposed to the atmosphere. Further work will show whether this time limit can be somewhat extended.