The local regularity conditions for the Navier–Stokes equations via one directional derivative of the velocity

Lithuanian Mathematical Journal - We study the local regularity of solutions to the Navier–Stokes equations. We show for a suitable weak solution (u, p) on an open space-time domain D that if...     

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.     

Properties, solution state behavior, and crystal structures of chelates of DOTMA

The chemistry of polyamino carboxylates and their use as ligands for Ln(3+) ions is of considerable interest from the point of view of the development of new imaging agents. Of particular interest is the chemistry of the macrocyclic ligand 1,4,7,10-tetraazacyclododecane-1,4,7,10-tetraacetic acid (DOTA) and its derivatives. Herein we report that the tetramethylated DOTA derivative, DOTMA, possess several properties that, from an imaging agent development point of view, are more advantageous than those of the parent DOTA. In particular, the Ln(3+) chelates of DOTMA exhibit a marked preference for the monocapped twisted square antiprismatic coordination isomer which imparts more rapid water exchange kinetics on the chelates; τ(M)(298) was determined to be 85 ns for GdDOTMA. Differential analysis of the (17)O R(2ρ) temperature profiles of both GdDOTA and GdDOTMA afforded the τ(M)(298) values for the square (SAP) and twisted square antiprismatic (TSAP) isomers of each chelate that were almost identical: 365 ns (SAP) and 52 ns (TSAP). The origin of this accelerated water exchange in the TSAP isomer appears to be the slightly longer Gd-OH(2) bond distance (2.50 ?) that is observed in the crystal structure of GdDOTMA which crystallizes in the P(2) space group as a TSAP isomer. The Ln(3+) chelates of DOTMA also exhibit high thermodynamic stabilities ranging from log K(ML) = 20.5 for CeDOTMA, 23.5 for EuDOTMA and YbDOTMA comparable to, but a shade lower than, those of DOTA.     

Turing incomparability in Scott sets

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