The Coordination Structure of the Extracted Cobalt(II) Complex with a Synergistic Mixture Containing Lix63 and Versatic10

In the present work, the cobalt(II ) synergist complex with isobutyric acid (HL^I ) and 5‐hydroxy‐4‐octanone oxime (HB^I ), which were the corresponding short‐chain analogs of active synergistic mixture of Versatic10 (HL ) and Lix63 (5,8‐diethyl‐7‐hydroxy‐6‐dodecanoneoxime, HB ), was prepared and studied by X‐ray single‐crystal diffraction. The crystal structure of the cobalt(II ) synergistic complex showed that the composition of the complex was Co(HB^I )₂(L^I )₂ with a cis‐form octahedron geometry structure. Both intra and intermolecular hydrogen bonding between the uncoordinated carbonyl oxygen atom of the deprotonated monodentate anionic ligand L^I and the hydrogen atom of the α‐hydroxy or the oxime hydroxyl group of HB^I were observed in the crystal lattice. In order to bridge the gap between the solid‐state structure of the cobalt(II ) synergist complex and the solution structure of the extracted cobalt(II ) complex with the actual synergistic mixture containing Versatic10 and Lix63 in the nonpolar organic phase, both the cobalt(II ) synergistic complex and the extracted cobalt(II ) complex were further investigated by Fourier transform infrared spectroscopy (FT‐IR ) and electrospray ionization mass spectrometry (ESI‐MS ). The results indicated that the extracted cobalt(II ) complex in the nonpolar organic phase might possess a similar coordination structure as that of the cobalt(II ) synergist complex.     

Ends,tangles and critical vertex sets

We show that an arbitrary infinite graph G can be compactified by its ends plus its critical vertex sets, where a finite set X of vertices of an infinite graph is critical if its deletion leaves some infinitely many components each with neighbourhood precisely equal to X. We further provide a concrete separation system whose ?₀‐tangles are precisely the ends plus critical vertex sets. Our tangle compactification is a quotient of Diestel's (denoted by ), and both use tangles to compactify a graph in much the same way as the ends of a locally finite and connected graph compactify it in its Freudenthal compactification. Finally, generalising both Diestel's construction of and our construction of , we show that G can be compactified by every inverse limit of compactifications of the sets of components obtained by deleting a finite set of vertices. Diestel's is the finest such compactification, and our is the coarsest one. Both coincide if and only if all tangles are ends. This answers two questions of Diestel.     

An algorithm for constrained convex optimization

This paper presents a feasible direction algorithm for the minimization of a pseudoconvex function over a smooth, compact, convex set. We establish that each cluster point of the generated sequence is an optimal solution of the problem without introducing anti-jamming procedures. Each iteration of the algorithm involves as subproblems only one line search for a zero of a continuously differentiable convex function and one univariate function minimization on a compact interval.     

Classes of 2-step nilpotent lie algebras

We give a characterization of the Lie algebras of H-type independent of the inner product used in the definition. We classify the real 2-step nilpotent Lie algebras with 2-dimensional center. Using these results we give examples of regular Lie algebras that are not H-type.     

Splitting families of sets in ZFC

Miller's 1937 splitting theorem was proved for every finite n>0$n>0$ for all ρ-uniform families of sets in which ρ is infinite. A simple method for proving Miller-type splitting theorems is presented here and an extension of Miller's theorem is proved in ZFC for every cardinal ν for all ρ  -uniform families in which ρ≥_?ω(ν)$\rho \ge {\mathrm{?}}_{\omega }(\nu )$. The main ingredient in the method is an asymptotic infinitary Löwenheim–Skolem theorem for anti-monotone set functions.     

Multi-variable weighted geometric means of positive definite matrices

We define a family of weighted geometric means {G(t;ω;A)}_t∈[0,1]ⁿ where ω and A vary over all positive probability vectors in Rⁿ and n-tuples of positive definite matrices resp. Each of these weighted geometric means interpolates between the weighted ALM (t=0_n) and BMP (t=1_n) geometric means (ALM and BMP geometric means have been defined by Ando-Li-Mathias and Bini-Meini-Poloni, respectively.) We show that the weighted geometric means satisfy multidimensional versions of all properties that one would expect for a two-variable weighted geometric mean.     

A tension approach to controlling the shape of cubic spline surfaces on FVS triangulations

We propose a parametric tensioned version of the FVS macro-element to control the shape of the composite surface and remove artificial oscillations, bumps and other undesired behaviour. In particular, this approach is applied to C¹ cubic spline surfaces over a four-directional mesh produced by two-stage scattered data fitting methods.     

On the theory of Pfaffians based on exponential maps in exterior algebras

Based on the relation of exponential maps and interior products in exterior algebras, some formulas of Pfaffians, including expansion formulas and the Cayley-Jacobi formula for determinants of alternating matrices, are deduced with new proofs. As an application, Pfaffian powers of alternating bilinear forms [O. Loos, Discriminant algebras and adjoints of quadratic forms, Beiträge Algebra Geom. 38 (1997) 33-72] are interpreted in terms of exponential maps in algebras of alternating multi-linear forms.     

Hermite interpolation with radial basis functions on spheres

We show how conditionally negative definite functions on spheres coupled with strictly completely monotone functions (or functions whose derivative is strictly completely monotone) can be used for Hermite interpolation. The classes of functions thus obtained have the advantage over the strictly positive definite functions studied in [17] that closed form representations (as opposed to series expansions) are readily available. Furthermore, our functions include the historically significant spherical multiquadrics. Numerical results are also presented. This revised version was published online in June 2006 with corrections to the Cover Date.     

A note on commutators of Hardy operators

Fu and Lu et al. 7 showed that the commutator $\mathcal {H}_{\beta ,b}$ generated by the fractional Hardy operator and a locally integrable function b is bounded on the homogenous Herz spaces if and only if b is a central bounded mean oscillation function. We show that their result is optimal by giving a counterexample.