On universal modules with pure embeddings

We show that certain classes of modules have universal models with respect to pure embeddings: Let R be a ring, T a first-order theory with an infinite model extending the theory of R-modules and ${\mathbf{K}}^T=(Mod(T),{\le }_{\mathrm{pp}})$ (where ⩽_pp stands for “pure submodule”). Assume ${\mathbf{K}}^T$ has the joint embedding and amalgamation properties. If ${\lambda }^{|T|}=\lambda$ or $\forall \mu <\lambda ({\mu }^{|T|}<\lambda )$, then ${\mathbf{K}}^T$ has a universal model of cardinality λ. As a special case, we get a recent result of Shelah [28, 1.2] concerning the existence of universal reduced torsion-free abelian groups with respect to pure embeddings. We begin the study of limit models for classes of R-modules with joint embedding and amalgamation. We show that limit models with chains of long cofinality are pure-injective and we characterize limit models with chains of countable cofinality. This can be used to answer [18, Question 4.25]. As this paper is aimed at model theorists and algebraists an effort was made to provide the background for both.     

Crystal structures and spectroscopic characterization of MBr2(CNXyl)n (M = Fe and Co,n = 4; M = Ni,n = 2; Xyl = 2,6‐dimethylphenyl), and of formally zero‐valent iron as a cocrystal of Fe(CNXyl)5 and Fe2(CNXyl)9

Structures and spectroscopic characterization of the divalent complexes cis‐dibromidotetrakis(2,6‐dimethylphenyl isocyanide)iron(II) dichloromethane 0.771‐solvate, [FeBr₂(C₉H₉N)₄]·0.771CH₂Cl₂ or cis‐FeBr₂(CNXyl)₄·0.771CH₂Cl₂ (Xyl = 2,6‐dimethylphenyl), trans‐dibromidotetrakis(2,6‐dimethylphenyl isocyanide)iron(II), [FeBr₂(C₉H₉N)₄] or trans‐FeBr₂(CNXyl)₄, trans‐dibromidotetrakis(2,6‐dimethylphenyl isocyanide)cobalt(II), [CoBr₂(C₉H₉N)₄] or trans‐CoBr₂(CNXyl)₄, and trans‐dibromidobis(2,6‐dimethylphenyl isocyanide)nickel(II), [NiBr₂(C₉H₉N)₂] or trans‐NiBr₂(CNXyl)₂, are presented. Additionally, crystals grown from a cold diethyl ether solution of zero‐valent Fe(CNXyl)₅ produced a structure containing a cocrystallization of mononuclear Fe(CNXyl)₅ and the previously unknown dinuclear [Fe(CNXyl)₃]₂(μ₂‐CNXyl)₃, namely pentakis(2,6‐dimethylphenyl isocyanide)iron(0) tris(μ₂‐2,6‐dimethylphenyl isocyanide)bis[tris(2,6‐dimethylphenyl isocyanide)iron(0)], [Fe(C₉H₉N)₅][Fe₂(C₉H₉N)₉]. The (M)C—N—C(Xyl) angles of the isocyanide ligand are nearly linear for the metals in the +2 oxidation state, for which the ligands function essentially as pure donors. The νCN stretching frequencies for these divalent metal isocyanides are at or above that of the free ligand. Relative to Fe^II, in the structure containing iron in the formally zero‐valent oxidation state, the Fe—C bond lengths have shortened, the C[triple‐bond]N bond lengths have elongated, the (M)C—N—C(Xyl) angles of the terminal CNXyl ligands are more bent, and the νCN stretching frequencies have shifted to lower energies, all indicative of substantial M(dπ)→π* backbonding.     

Do columnar defects produce bulk pinning?

From magneto-optical imaging performed on heavy-ion-irradiated YBa(2)Cu(3)O(7-delta) single crystals, it is found that at fields and temperatures where strong single vortex pinning by individual irradiation-induced amorphous columnar defects is to be expected, vortex motion is limited by the nucleation of vortex kinks at the specimen surface. In the material bulk, vortex motion occurs through (easy) kink sliding. Depinning in the bulk determines the screening current only at fields comparable to or larger than the matching field, at which the majority of vortices is not trapped by an ion track.     

Turing incomparability in Scott sets

For every Scott set and every nonrecursive set in , there is a such that and are Turing incomparable.

     

Crystal Structures of DOTMA Chelates from Ce3+ to Yb3+: Evidence for a Continuum of Metal Ion Hydration States

The crystal structures of chelates formed between each stable paramagnetic lanthanide ion and the octadentate polyamino carboxylate ligand DOTMA are described. A total of 23 individual chelates structures were obtained; in each chelate the coordination geometry around the metal ion is best described as a twisted square antiprism (torsion angle −25.0°–−31.4°). Despite the uniformity of the general coordination geometry provided by the DOTMA ligand, there is a considerable variation in the hydration state of each chelate. The early Ln³⁺ chelates are associated with a single inner sphere water molecule; the Ln-OH₂ interaction is remarkable for being very long. After a clear break at gadolinium, the number of chelates in the unit cell that have a water molecule interacting with the Ln³⁺ decreases linearly until at Tm³⁺ no water is found to interact with the metal ion. The Ln-OH₂ distance observed in the chelates of the later Ln³⁺ ions are also extremely long and increase as the ions contract (2.550–2.732 Å). No clear break between hydrated and dehydrated chelates is observed; rather this series of chelates appear to represent a continuum of hydration states in which the ligand gradually closes around the metal ion as its ionic radius decreases (with decreased hydration) and the metal drops down into the coordination cage.     

Generation of dimethylsilylene and allylidene holmium complexes from trimethylsilylated allyl ligands

The reaction of K[1,3-(SiMe3)2C3H3] with partially hydrated holmium triflate leads to a dimeric complex (1) in which hydrogen abstraction from a trimethylsilyl group has occurred on two allyl ligands, forming dimethylsilylene units that bridge the holmium atoms. When the reaction time is prolonged, a different product (2) is isolated, in which in addition to two dimethylsilylene bridges, the metal centers are joined with a mu-eta1,eta3-allylidene ligand. Both crystallographic and computational studies provide evidence for delocalized bonding in the allylidene fragment.