Differential Algebras with Banach-Algebra Coefficients II: The Operator Cross-Ratio Tau-Function and the Schwarzian Derivative

Several features of an analytic (infinite-dimensional) Grassmannian of (commensurable) subspaces of a Hilbert space were developed in the context of integrable PDEs (KP hierarchy). We extended some of those features when polarized separable Hilbert spaces are generalized to a class of polarized Hilbert modules and then consider the classical Baker and τ-functions as operator-valued. Following from Part I we produce a pre-determinant structure for a class of τ-functions defined in the setting of the similarity class of projections of a certain Banach *-algebra. This structure is explicitly derived from the transition map of a corresponding principal bundle. The determinant of this map leads to an operator τ-function. We extend to this setting the operator cross-ratio which had previously been used to produce the scalar-valued τ-function, as well as the associated notion of a Schwarzian derivative along curves inside the space of similarity classes of a given projection. We link directly this cross-ratio with Fay’s trisecant identity for the τ-function. By restriction to the image of the Krichever map, we use the Schwarzian to introduce the notion of an operator-valued projective structure on a compact Riemann surface: this allows a deformation inside the Grassmannian (as it varies its complex structure). Lastly, we use our identification of the Jacobian of the Riemann surface in terms of extensions of the Burchnall–Chaundy C*-algebra (Part I) provides a link to the study of the KP hierarchy.     

The Robin problem for the scalar Oseen equation

We study the Robin problem for the scalar Oseen equation in an open n‐dimensional set with compact Ljapunov boundary. We prescribe two types of Robin boundary conditions, and prove the unique solvability of these problems as well as a representation formula for the solution in form of a scalar Oseen single layer potential. Moreover, we prove the maximum principle for the solution to the Robin problem of the scalar Oseen equation. Copyright © 2013 John Wiley & Sons, Ltd.     

Rapid Assembly of Saturated Nitrogen Heterocycles in One‐Pot: Diazo‐Heterocycle “Stitching” by N–H Insertion and Cyclization

Methods that provide rapid access to new heterocyclic structures in biologically relevant chemical space provide important opportunities in drug discovery. Here, a strategy is described for the preparation of 2,2‐disubstituted azetidines, pyrrolidines, piperidines, and azepanes bearing ester and diverse aryl substituents. A one‐pot rhodium catalyzed N–H insertion and cyclization sequence uses diazo compounds to stitch together linear 1,m‐haloamines (m=2–5) to rapidly assemble 4 ‐, 5 ‐, 6 ‐, and 7 ‐membered saturated nitrogen heterocycles in excellent yields. Over fifty examples are demonstrated, including examples with diazo compounds derived from biologically active compounds. The products can be functionalized to afford α,α‐disubstituted amino acids and applied to fragment synthesis.     

Solubility Thermodynamics of CyMe$$_{}$$-BTBP in Various Diluents Mixed with TBP

The two organic ligands 6,6′-bis(5,5,8,8-tetramethyl-5,6,7,8-tetrahydrobenzo[1,2,4]triazin-3-yl)[2,2′]bipyridine (CyMe\(_{4}\)-BTBP) and tri-butyl phosphate (TBP) have previously been investigated in different diluents for use within recycling of used nuclear fuel through solvent extraction. The thermodynamic parameters, \(K_{\mathrm{S}}\), \(\Delta C_{p}\), \(\Delta H^{0}\) and \(\Delta S^{0}\), of the CyMe\(_{4}\)-BTBP solubility in three diluents (cyclohexanone, octanol and phenyl trifluoromethyl sulfone) mixed with TBP have been studied at 288, 298 and 308 K, both as pristine solutions and pre-equilibrated with 4 mol\(\cdot \)L\(^{-1}\) nitric acid. In addition, the amount of acid in the organic phase and density change after pre-equilibration have been measured. The solubility of CyMe\(_{4}\)-BTBP increases with an increased temperature in all systems, especially after acid pre-equilibration. This increased CyMe\(_{4}\)-BTBP solubility after pre-equilibration could be explained by acid dissolution into the solvent. Comparing the \(\Delta H^{0}\) and \(\Delta S^{0}\) calculated using \(\Delta C_{p}\) with the same parameters derived from a linear fit indicates temperature independence of all three thermodynamic systems. The change in enthalpy is positive in all solutions.     

On the application of ICP-MS techniques for measuring uranium and plutonium: a Nordic inter-laboratory comparison exercise

Inductively coupled plasma mass spectrometry (ICP-MS) techniques are widely used for determination of long-lived radionuclides and their isotopic ratios in the nuclear fields. Uranium (U) and Plutonium (Pu) isotopes have been determined by many researchers with ICP-MS due to its relatively high sensitivity and short measurement time. In this work, an inter-laboratory comparison exercise among the Nordic countries was performed, focusing on the measurement of U and Pu isotopes in certified reference materials by ICP-MS. The performance and characters of different ICP-MS instruments are evaluated and discussed in this paper.     

Comparing results of X-ray diffraction, µ-Raman spectroscopy and neutron diffraction when identifying chemical phases in seized nuclear material,during a comparative nuclear forensics exercise

Journal of Radioanalytical and Nuclear Chemistry - This work presents the results for identification of chemical phases obtained by several laboratories as a part of an international nuclear...     

Total Synthesis of Tambromycin by Combining Chemocatalytic and Biocatalytic C−H Functionalization

A combination of genomic and metabolomic approaches recently resulted in the identification of a nonribosomal tetrapeptide tambromycin, which possesses promising antiproliferative activity and several unusual structural features, including a densely substituted indole, a methyloxazoline ring, and an unusual pyrrolidine‐containing amino acid called tambroline. In this work, we identify a concise synthetic route to access tambromycin, which relies on the strategic use of biocatalytic and chemocatalytic C?H functionalization methods to prepare two key precursors to the natural product in an efficient and scalable manner. The success of our study highlights the benefits of applying the principles of biocatalytic retrosynthesis as well as C?H functionalization logic to the synthesis of complex molecular scaffolds.