On Ricci solitons of cohomogeneity one

We analyse some properties of the cohomogeneity one Ricci soliton equations, and use Ansätze of cohomogeneity one to produce new explicit examples of complete Kähler Ricci solitons of expanding, steady and shrinking types. These solitons are foliated by hypersurfaces which are circle bundles over a product of Fano Kähler–Einstein manifolds or over coadjoint orbits of a compactly connected semisimple Lie group.     

Optimal strong Mal’cev conditions for omitting type 1 in locally finite varieties

We show that the class of locally finite varieties omitting type 1 has the following properties. This class is:

1. definable by an idempotent, linear, strong Mal’cev condition in a language with one 4-ary function symbol;
2. not definable by an idempotent, linear, strong Mal’cev condition in a language with only one function symbol of arity strictly less than 4;
3. definable by an idempotent, linear, strong Mal’cev condition in a language with two 3-ary function symbols;
4. not definable by an idempotent, linear, strong Mal’cev condition in a language with function symbols of arity less than 4 unless at least two of the symbols have arity 3.

     

NO Reactions Inside Crystals

Reactions of chemisorbed reagents inside the crystalline molecular solid state are rare but offer unexploited methods for selective solvent-free chemical synthesis. Here we show that the greenhouse gas precursor, nitric oxide (NO) is chemisorbed by crystals of the hexafluorophosphate salts of complexes containing dicobalt sites. On NO sorption a cascade of reactions results in the in-crystal synthesis of nitrite and other gaseous NO_x. Recrystallization enabled structural elucidation of the mixed valent {[(bpbp)Co₂(μ-(η¹-O : η¹-N)-ONO)]₂(bdc)}⁴⁺ (bpbp=2,6-bis(N,N-bis(2-pyridylmethyl)aminomethyl)-4-tert-butylphenolato, bdc=1,4-benzenedicarboxylato) cation. Overlapping signals in the solid-state EPR spectra confirm the Co^IICo^III oxidation state and the presence of NO₂ trapped inside the unrecrystallised solid products (br. g=4, triplet g=2 (340 mT), A(N)=73 MHz), despite three cycles of vacuum and N₂ flushing. Consistently, ν_N−O bands appear in the Raman and IR spectra that are due to the coordinated nitrate and the trapped NO₂ that were synthesized in-crystal. The latter is expelled by heating the solid to 160 °C or by recrystallization. Dimetallic cooperativity is proposed for the NO transformations in these rare examples of selective, chemisorptive substrate reactions in the solid-state.