The tensor Golub–Kahan–Tikhonov method applied to the solution of ill-posed problems with a t-product structure

This paper discusses an application of partial tensor Golub–Kahan bidiagonalization to the solution of large-scale linear discrete ill-posed problems based on the t-product formalism for third-order tensors proposed by Kilmer and Martin (M. E. Kilmer and C. D. Martin, Factorization strategies for third order tensors, Linear Algebra Appl., 435 (2011), pp. 641-658). The solution methods presented first reduce a given (large-scale) problem to a problem of small size by application of a few steps of tensor Golub–Kahan bidiagonalization and then regularize the reduced problem by Tikhonov's method. The regularization operator is a third-order tensor, and the data may be represented by a matrix, that is, a tensor slice, or by a general third-order tensor. A regularization parameter is determined by the discrepancy principle. This results in fully automatic solution methods that neither require a user to choose the number of bidiagonalization steps nor the regularization parameter. The methods presented extend available methods for the solution for linear discrete ill-posed problems defined by a matrix operator to linear discrete ill-posed problems defined by a third-order tensor operator. An interlacing property of singular tubes for third-order tensors is shown and applied. Several algorithms are presented. Computed examples illustrate the advantage of the tensor t-product approach, in comparison with solution methods that are based on matricization of the tensor equation.     

GCV for Tikhonov regularization via global Golub–Kahan decomposition

Generalized cross validation is a popular approach to determining the regularization parameter in Tikhonov regularization. The regularization parameter is chosen by minimizing an expression, which is easy to evaluate for small‐scale problems, but prohibitively expensive to compute for large‐scale ones. This paper describes a novel method, based on Gauss‐type quadrature, for determining upper and lower bounds for the desired expression. These bounds are used to determine the regularization parameter for large‐scale problems. Computed examples illustrate the performance of the proposed method and demonstrate its competitiveness. Copyright © 2016 John Wiley & Sons, Ltd.     

On the strong Brillinger-mixing property of α-determinantal point processes and some applications

First, we derive a representation formula for all cumulant density functions in terms of the non-negative definite kernel function C(x, y) defining an α-determinantal point process (DPP). Assuming absolute integrability of the function C₀(x) = C(o, x), we show that a stationary α-DPP with kernel function C₀(x) is “strongly” Brillinger-mixing, implying, among others, that its tail-σ-field is trivial. Second, we use this mixing property to prove rates of normal convergence for shot-noise processes and sketch some applications to statistical second-order analysis of α-DPPs.     

Absolute Configuration from Different Multifragmentation Pathways in Light‐Induced Coulomb Explosion Imaging

The absolute configuration of individual small molecules in the gas phase can be determined directly by light‐induced Coulomb explosion imaging (CEI). Herein, this approach is demonstrated for ionization with a single X‐ray photon from a synchrotron light source, leading to enhanced efficiency and faster fragmentation as compared to previous experiments with a femtosecond laser. In addition, it is shown that even incomplete fragmentation pathways of individual molecules from a racemic CHBrClF sample can give access to the absolute configuration in CEI. This leads to a significant increase of the applicability of the method as compared to the previously reported complete break‐up into atomic ions and can pave the way for routine stereochemical analysis of larger chiral molecules by light‐induced CEI.     

Coupling techniques in speciation analysis

Summary Direct coupling techniques in speciation analysis lead to reproducible results with lower risks of trace contamination or losses of analytes in a shorter period of time. The direct coupling mode seems to be most promising for liquid chromatographic separation methods — especially gel permeation chromatography — and for some electrophoretic methods — as flow-through and capillary techniques. Promising detection methods are Flame-AAS and ICP-OES but also hydride-generation AAS and chemical reaction detectors in continuous-flow technique.     

Dimethylformamide-Exchange Mechanism on Some First-Row Divalent Transition-Metal Ions

The effect of temperature on the dimethylformamide exchange on Mn(DMF) and Fe(DMF) has been studied by ¹³C- and ¹⁷O-NMR, respectively, yielding the following kinetic parameters: k²⁹⁸ equals; (2.2±0.2). 10⁶ S^?1, ΔH^≠ = 34.6 ± 1.3 kJ mol^?1, ΔS^≠ = ?7.4 ± 4.8 J K^?1mol^?1 for Mn²⁺ and K²⁹⁸ = (9.7 ± 0.2).10⁵ S^?1, Delta;H^≠ = 43.0 ± 0.9 kJ mol^?1, ΔS^≠ = + 13.8 ± 2.8 J K^?1mol^?1 for Fe²⁺. The volumes of activation, ΔV^≠ in cm³mol^?1, derived from high-pressure NMR on these metal ions, together with the previously published activation volumes for Co²⁺ and Ni²⁺ (+2.4 ± 0.2 (Mn²⁺), +8.5 ± 0.4 (Fe²⁺) +9.2 ± 0.3 (Co²⁺), + 9.1 ± 0.3 (Ni²⁺)) give evidence for a dissociative activation mode for DMF exchange on these high-spin first-row transition-metal divalent ions. The small positive ΔV^≠ value observed for DMF exchange on Mn²⁺ seems to indicate that a mechanistic changeover also occurs along the series, (probably from I_d to D), as for the other solvents previously studied (I_a to I_d, for H₂O, MeOH, MeCN). This changeover is shifted to the earlier elements of the series, due to more pronounced steric crowding for dimethylformamide hexasolvates.     

Water Exchange on Hexaaquavanadium(III): a Variable-Temperature and Variable-Pressure 17O-NMR Study at 1.4 and 4.7 Tesla

Water exchange on hexaaquavanadium (III) has been studied as a function of temperature (255 to 413 K) and pressure (up to 250 MPa, at several temperatures) by ¹⁷O-NMR spectroscopy at 8.13 and 27.11 MHz. The samples contained V³⁺ (0.30–1.53 m), H⁺ (0.19–2.25 m) and ¹⁷O-enriched (10–20%) H₂O. The trifluoromethanesulfonate was used as counter-ion, and, contrary to the previously used chloride or bromide, CF₃SO is shown to be non-coordinating. The following exchange parameters were obtained: k = (5.0 ± 0.3) · 10² s^?1, ΔH* = (49.4 ± 0.8) kJ mol^?1, ΔS* = ?(28 ± 2) JK^?1 mol^?1, ΔV* = ?(8.9 ± 0.4) cm³ mol^?1 and Δβ* = ?(1.1 ± 0.3) · 10^?2 cm³ mol^?1 MPa^?1. They are in accord with an associative interchange mechanism, I_a. These results for H₂O exchange are discussed together with the available data for complex formation reactions on hexaaquavanadium(III). A semi-quantitative analysis of the bound H₂O linewidth led to an estimation of the proportions of the different contributions to the relaxation mechanism in the coordinated site: the dipole-dipole interaction hardly contributes to the relaxation (less than 7%); the quadrupolar relaxation, and the scalar coupling mechanism are nearly equally efficient at low temperature (～ 273 K), but the latter becomes more important at higher temperature (75–85% contribution at 360 K).     

Cyclopalladation of 1,1′-Azonaphthalene (= Di(naphthalen-1-yl)diazene): Isolation and characterization of two isomeric complexes

The cyclopalladation of 1,1′-azonaphthalene (= di(naphthalen-1-yl)diazene; 2 ) with bis(hexafluoroacetyl-acetonato)palladium(II) (3; [Pd(hfa)₂]) yields the ortho-palladated complex (1,1,1,5,5,5-hexafluoropentane-2,4-dionato-κ²O,O′)[1-(naphthalen-1-ylazo-κN²)naphthalen-2-yl-κC²]palladium(II) ( 4 ) as well as the peri-palladated complex (1,1,1,5,5,5-hexafluoropentane-2,4-dionato-κ²O,O′)[8-(naphthalen-1-ylazo-κN²)naphthalen-1-yl-κC¹]-palladium(II) ( 5 ); their structures were corroborated by X-ray analyses. The formation of the novel peri-metallated product 5 containing a six-membered palladacycle strongly depends upon the reaction conditions.