On pointwise decay rates of time‐periodic solutions to the Navier–Stokes equation

We study the existence of a time‐periodic solution with pointwise decay properties to the Navier–Stokes equation in the whole space. We show that if the time‐periodic external force is sufficiently small in an appropriate sense, then there exists a time‐periodic solution $\{u,p\}$ of the Navier–Stokes equation such that $|{?}^j{u(t,x)|=O(|x|}^{1?n?j})$ and $|{?}^j{p(t,x)|=O(|x|}^{?n?j})(j=0,1,\dots )$ uniformly in $t\in \mathbb{R}$ as $|x|\to \infty$. Our solution decays faster than the time‐periodic Stokes fundamental solution and the faster decay of its spatial derivatives of higher order is also described.     

On Default Priors for Robust Bayesian Estimation with Divergences

This paper presents objective priors for robust Bayesian estimation against outliers based on divergences. The minimum γ-divergence estimator is well-known to work well in estimation against heavy contamination. The robust Bayesian methods by using quasi-posterior distributions based on divergences have been also proposed in recent years. In the objective Bayesian framework, the selection of default prior distributions under such quasi-posterior distributions is an important problem. In this study, we provide some properties of reference and moment matching priors under the quasi-posterior distribution based on the γ-divergence. In particular, we show that the proposed priors are approximately robust under the condition on the contamination distribution without assuming any conditions on the contamination ratio. Some simulation studies are also presented.     

Defect-Free Synthesis of a Fully π-Conjugated Helical Ladder Polymer and Resolution into a Pair of Enantiomeric Helical Ladders**

Fully π-conjugated ladder polymers with a spiral geometry represent a new class of helical polymers with great potential for organic nanodevices, but there is no precedent for an optically active helical ladder polymer totally composed of achiral units. We now report the defect-free synthesis and resolution of a fully π-conjugated helical ladder polymer with a rigid helical cavity, which has been achieved by quantitative and chemoselective acid-promoted alkyne benzannulations of a rationally designed, random-coil achiral polymer followed by chromatographic enantioseparation. Because of a sufficiently high helix-inversion barrier, the isolated excess one-handed helical ladder polymer with a degree of polymerization of more than 15 showed a strong circular dichroism with a dissymmetry factor of up to 1.7×10⁻² and is thermally stable, maintaining its optical activity in solution even at 100 °C, as well-supported by molecular dynamics simulation.     

Unique Thermal Structural Phase Transitions Exhibited by Unsymmetrical Organometallic Gold(III)-Dithiolene Complexes with Pentylthio and Hexylthio Groups

Unsymmetrical gold(III)-dithiolene complexes are potential candidates for molecular materials that exhibit thermal structural phase transitions. In this study, unsymmetrical ppy-gold(III) (ppy⁻=C-deprotonated-2-phenylpyridine(−)) complexes [AuC5] and [AuC6] coordinated by dithiolene ligands containing tetrathiafulvalene (TTF) skeletons with pentylthio (2-{bis(pentylthio)-1,3-dithiol-2-ylidene}-1,3-dithiol-4,5-dithiolate(2−)) and hexylthio groups (2-{bis(hexylthio)-1,3-dithiol-2-ylidene}-1,3-dithiol-4,5-dithiolate(2−)) were synthesized. Both complexes exhibited a large absorption band at approximately 508 nm, owing to intramolecular ligand-to-ligand charge transfer. One-dimensional columnar structures with head-to-tail molecular arrangements around the metal ions were constructed in the crystals. The flexible alkylthio groups were intercalated into crystalline spaces between dithiolene ligands in the columns. [AuC5] exhibits a simple phase transition at 198 °C between crystalline and isotropic phases irreversibly. The crystalline phase of [AuC6] observed at 25 °C melted at 148 °C. Another crystalline phase grew above 148 °C with a very slow crystallization rate from the liquid phase and was completely transformed into an isotropic phase at 200 °C.     

Persistent Dialkylsilanone Generated by Dehydrobromination of Dialkylbromosilanol

A persistent dialkylsilanone was synthesized by the dehydrobromination of a dialkylbromosilanol with tris(trimethylsilyl)silyl potassium in solution at ?80 °C: It was characterized by NMR and IR spectroscopy, and was tested in several reactions. In ²⁹Si NMR spectrum in [D₈]toluene, the signal due to the unsaturated silicon nuclei was observed at 128.7 ppm. Reactions of the dialkylsilanone with water and mesitonitrile oxide gave a silanediol and a [2+3] cycloadduct, respectively. The silanone remains intact in [D₈]toluene below ?80 °C for at least two days, while it undergoes unprecedented isomerization to give a siloxysilene by means of 1,3‐silyl migration at higher temperatures.     

Reconstitution of Biosynthetic Machinery for the Synthesis of the Highly Elaborated Indole Diterpene Penitrem

Penitrem A is one of the most elaborated members of the fungal indole diterpenes. Two separate penitrem gene clusters were identified using genomic and RNA sequencing data, and 13 out of 17 transformations in the penitrem biosynthesis were elucidated by heterologous reconstitution of the relevant genes. These reactions involve 1) a prenylation‐initiated cationic cyclization to install the bicyclo[3.2.0]heptane skeleton (PtmE), 2) a two‐step P450‐catalyzed oxidative processes forming the unique tricyclic penitrem skeleton (PtmK and PtmU), and 3) five sequential oxidative transformations (PtmKULNJ). Importantly, without conventional gene disruption, reconstitution of the biosynthetic machinery provided sufficient data to determine the pathway. It was thus demonstrated that the Aspergillus oryzae reconstitution system is a powerful method for studying the biosynthesis of complex natural products.     

Specific reactivity of 2,4,6-tri-tert-butylanilide anions and its application to benzylation reagent

The reaction of methyl iodide with an anilide anion prepared from 2,4,6-tri-tert-butylanilide and NaH in CH₃CN gave N-methyl anilide (N-alkylation product) as a major product, while in the reaction of benzyl bromide with the anilide anion in DMF, O-benzyl imidate (O-alkylation product) was obtained with almost complete selectivity. The treatment of O-benzyl imidate with alcohols and carboxylic acids in the presence of trifluoromethane sulfonic acid gave benzyl ethers and benzyl esters, respectively.     

Proton Conduction at High Temperature in High-Symmetry Hydrogen-Bonded Molecular Crystals of RuIII Complexes with Six Imidazole-Imidazolate Ligands

A new H-bonded crystal [Ru^III(Him)₃(Im)₃] with three imidazole (Him) and three imidazolate (Im⁻) groups was prepared to obtain a higher-temperature proton conductor than a Nafion membrane with water driving. The crystal is constructed by complementary N−H⋅⋅⋅N H-bonds between the Ru^III complexes and has a rare Icy-c* cubic network topology with a twofold interpenetration without crystal anisotropy. The crystals show a proton conductivity of 3.08×10⁻⁵ S cm⁻¹ at 450 K and a faster conductivity than those formed by only HIms. The high proton conductivity is attributed to not only molecular rotations and hopping motions of HIm frameworks that are activated at ∼113 K, but also isotropic whole-molecule rotation of [Ru^III(Him)₃(Im)₃] at temperatures greater than 420 K. The latter rotation was confirmed by solid-state ²H NMR spectroscopy; probable proton conduction routes were predicted and theoretically considered.