Absence of Anomalous Dissipation of Energy in Forced Two Dimensional Fluid Equations

We prove the absence of anomalous dissipation of energy for long time averaged solutions of the forced critical surface quasi-geostrophic equation in two spatial dimensions.     

Holomorphic maps preserving parts of the local spectrum

Let x ₀ be a nonzero vector in \({\mathbb{C}^{n}}\) , and let \({U\subseteq \mathcal{M}_{n}}\) be a domain containing the zero matrix. We prove that if φ is a holomorphic map from U into \({\mathcal{M}_{n}}\) such that the local spectrum of T ∈ U at x ₀ and the local spectrum of φ(T) at x ₀ have always a common value, then T and φ(T) have always the same spectrum, and they have the same local spectrum at x ₀ a.e. with respect to the Lebesgue measure on U. If \({\varphi \colon U\rightarrow \mathcal{M}_{n}}\) is holomorphic with φ(0) = 0 such that the local spectral radius of T at x ₀ equals the local spectral radius of φ(T) at x ₀ for all T ∈ U, there exists \({\xi \in \mathbb{C}}\) of modulus one such that ξT and φ(T) have the same spectrum for all T in U. We also prove that if for all T ∈ U the local spectral radius of φ(T) coincides with the local spectral radius of T at each vector x, there exists \({\xi \in \mathbb{C}}\) of modulus one such that φ(T) = ξT on U.     

Gadolinium Photocatalysis: Dearomative [2+2] Cycloaddition/Ring‐Expansion Sequence with Indoles

Lanthanide photocatalysts are much less investigated in synthetic chemistry than rare and expensive late transition metals. We herein introduce Gd^III photocatalysis of a highly regioselective, intermolecular [2+2] photocycloaddition/ring‐expansion sequence with indoles, which could provide divergent access to cyclopenta[b]indoles and indolines. A simple and commercially available Gd(OTf)₃ salt is sufficient for this visible‐violet‐light‐induced transformation. The reaction proceeds either through a transient or start‐to‐end dearomatization cascade and shows excellent regioselectivity (usually >95:5 r.r.), broad scope (59 examples), good functional group tolerance and facile scale‐up under mild, direct visible‐light‐excitation conditions. Mechanistic investigations reveal that direct excitation of the Gd(OTf)₃/indole mixture gives an excited state intermediate, which undergoes the subsequent [2+2] cycloaddition and cyclobutane‐expansion cascade.     

The [(NHC)B(H)C6F5]+ Cations and Their [B](H)−CO Borane Carbonyls

Hydride abstraction from the heterocyclic carbene borane adducts (NHC)BH₂C₆F₅ (NHC: IMes or IMe₄) gave the B−H containing [(NHC)B(H)C₆F₅]⁺ borenium cations. They added carbon monoxide to give the respective [(NHC)B(H)(C₆F₅)CO]⁺ boron carbonyl cations. Carbon nucleophiles add to the boron carbonyl to give [B](H) acyls. Hydride reduced the [B]CO cation to hydroxymethylborane derivatives.     

Light-Activated Carbon Monoxide Prodrugs Based on Bipyridyl Dicarbonyl Ruthenium(II) Complexes

Two photoactivatable dicarbonyl ruthenium(II) complexes based on an amide-functionalised bipyridine scaffold (4-position) equipped with an alkyne functionality or a green-fluorescent BODIPY (boron-dipyrromethene) dye have been prepared and used to investigate their light-induced decarbonylation. UV/Vis, FTIR and ¹³C NMR spectroscopies as well as gas chromatography and multivariate curve resolution alternating least-squares analysis (MCR-ALS) were used to elucidate the mechanism of the decarbonylation process. Release of the first CO molecule occurs very quickly, while release of the second CO molecule proceeds more slowly. In vitro studies using two cell lines A431 (human squamous carcinoma) and HEK293 (human embryonic kidney cells) have been carried out in order to characterise the anti-proliferative and anti-apoptotic activities. The BODIPY-labelled compound allows for monitoring the cellular uptake, showing fast internalisation kinetics and accumulation at the endoplasmic reticulum and mitochondria.     

Flow due to a plate that applies an accelerated shear to a second grade fluid between two parallel walls perpendicular to the plate

The velocity field and the shear stresses corresponding to the motion of a second grade fluid between two side walls, induced by an infinite plate that applies an accelerated shear stress to the fluid, are determined by means of the integral transforms. The obtained solutions, presented under integral form in term of the solutions corresponding to the flow due to a constant shear on the boundary, satisfy all imposed initial and boundary conditions. In the absence of the side walls, they reduce to the similar solutions over an infinite plate. The Newtonian solutions are obtained as limiting cases of the general solutions. The influence of the side walls on the fluid motion as well as a comparison between the two models is shown by graphical illustrations.