A multigrid perspective on the parallel full approximation scheme in space and time

For the numerical solution of time‐dependent partial differential equations, time‐parallel methods have recently been shown to provide a promising way to extend prevailing strong‐scaling limits of numerical codes. One of the most complex methods in this field is the “Parallel Full Approximation Scheme in Space and Time” (PFASST). PFASST already shows promising results for many use cases and benchmarks. However, a solid and reliable mathematical foundation is still missing. We show that, under certain assumptions, the PFASST algorithm can be conveniently and rigorously described as a multigrid‐in‐time method. Following this equivalence, first steps towards a comprehensive analysis of PFASST using blockwise local Fourier analysis are taken. The theoretical results are applied to examples of diffusive and advective type.     

On the gap problem for the sequence ∥mβ∥

We discuss the gap problem for the sequence m used in our previous Letter (D. H. Mayer, Lett. Math. Phys. 16, 139–143 (1988)).     

Electrochemical Reduction of Arylxenon Tetrafluoroborates

Reduction potentials of arylxenon(II) tetrafluoroborates have been measured by cyclic voltammetry method on platinum disc and ultramicro electrodes. The electrochemical properties of the arylxenon(II) tetrafluoroborates are quite different to those of corresponding arylperfluoralkyliodine(III) salts.     

Synthesis and NMR Analysis in Solution of Oligo(3‐hydroxyalkanoic acid) Derivatives with the Side Chains of Alanine,Valine, and Leucine (β‐Depsides): Coming Full Circle from PHB to β‐Peptides to PHB

Oligomers of 3‐hydroxyalkanoic acids that contain two, three, and six residues with and without O‐terminal (tBu)Ph₂Si and C‐terminal PhCH₂ protection have been synthesized in such a way that the side chains on the oligoester backbone were those of the proteinogenic amino acids Ala (Me), Val (CHMe₂), and Leu (CH₂CHMe₂). The enantiomerically pure 3‐hydroxyalkanoates were obtained by Noyori hydrogenation of the corresponding 3‐oxo‐alkanoates with [Ru((R)‐binap)Cl₂](binap=2,2′bis(diphenylphosphanyl)‐1,1′‐binaphthalene)/H₂ (Scheme 1), and the coupling was achieved under the conditions (pyridine/(COCl)₂, CH₂Cl₂, −78°) previously employed for the synthesis of various oligo(3‐hydroxybutanoic acids) (Schemes 2 and 3). The Cotton effects in the CD spectra of the new oligoesters provided no hints about chiral conformation (cf. a helix) in MeOH, MeCN, octan‐1‐ol, or CF₃CH₂OH solutions (Figs. 1 and 2). Detailed NMR investigations in CDCl₃ solution (Figs. 3–6, and Tables 1–5) of the hexa(3‐hydroxyalkanoic acid) with the side chains of Val (HC), Ala (HB), Leu (HH), Val, Ala, Leu (from O‐ to C‐terminus; 3 ) gave, on the NMR time‐scale, no evidence for the presence of any significant amount of a 2₁‐ or a 3₁‐helical conformation, comparable to those identified in stretched fibers of poly[(R)‐3‐hydroxybutanoic acid], or in lamellar crystallites and in single crystals of linear and cyclic oligo[(R)‐3‐hydroxybutanoic acids], or in the corresponding β‐peptide(s) (the oligo(3‐aminoalkanoic acid) analogs; 1 – 3 ). Thus, the extremely high flexibility (averaged or ‘random‐coil' conformation) of the polyester chain (CO−O rotational barrier ca. 13 kcal/mol; no hydrogen bonding), as compared to polyamide chains (CO−NH barrier ca. 18 kcal/mol; hydrogen bonding) has been demonstrated once again. The possible importance of this structural flexibility, which goes along with amphiphilic properties, for the role of PHB in biology, in evolution, and in prebiotic chemistry is discussed. Structural similarities of natural potassium‐channeling proteins and complexes of oligo(3‐hydroxybutanoates) with Na⁺, K⁺, or Ba²⁺ are alluded to (Figs. 7–9).     

Preparation of β2‐Homotryptophan Derivatives for β‐Peptide Synthesis

In view of the prominent role of the 1H‐indol‐3‐yl side chain of tryptophan in peptides and proteins, it is important to have the appropriately protected homologs H‐β² HTrp OH and H‐β³ HTrp OH (Fig.) available for incorporation in β‐peptides. The β²‐HTrp building block is especially important, because β²‐amino acid residues cause β‐peptide chains to fold to the unusual 12/10 helix or to a hairpin turn. The preparation of Fmoc and Z β²‐HTrp(Boc) OH by Curtius degradation (Scheme 1) of a succinic acid derivative is described (Schemes 2–4). To this end, the (S)‐4‐isopropyl‐3‐[(N‐Boc‐indol‐3‐yl)propionyl]‐1,3‐oxazolidin‐2‐one enolate is alkylated with Br CH₂CO₂Bn (Scheme 3). Subsequent hydrogenolysis, Curtius degradation, and removal of the Evans auxiliary group gives the desired derivatives of (R)‐H β²‐HTrp OH (Scheme 4). Since the (R)‐form of the auxiliary is also available, access to (S)‐β²‐HTrp‐containing β‐peptides is provided as well.     

Palladium-catalyzed, stereoselective rearrangement of a tetracyclic allyl cyclopropane under arylation

The first example of a pi,sigma domino-Heck reaction under concomitant rearrangement of the tetracyclic allylcyclopropane endo,exo-bishomobarrelene (5) is reported; the stereoselective reaction proceeds via an intramolecular insertion of a primarily-formed carbopalladation intermediate into a strained cyclopropane C-C sigma-bond, giving 9.     

Mechanical Properties of Bacterially Synthesized Nanocellulose Hydrogels

The mechanical characteristics of bacterially synthesized nano-cellulose (BNC) were studied with uniaxial compression and tensile tests. Compressive loads result in a release of water and the deformation of the water-saturated network corresponds approximately to the volume of released water. The BNC hydrogel exhibits a mainly viscous response under compression. The strain response under tensile loads has an elastic and a viscous component. This can be described with a Maxwell model, where the viscosity is strain rate-dependent. When the aqueous phase of the BNC hydrogel is stabilized with an additional alginate hydrogel matrix, the system exhibits an elastic response under compressive loads. The analysis of the ‘alginated’ BNC network with the Maxwell model shows that the alginate matrix increases the viscosity of the composite system. The results of the mechanical tests show that the water absorbed in the BNC hydrogel strongly influences its mechanical behavior.