Rhodium(I)‐Catalyzed Decarbonylative Spirocyclization through C?C Bond Cleavage of Benzocyclobutenones: An Efficient Approach to Functionalized Spirocycles

The rhodium‐catalyzed formation of all‐carbon spirocenters involves a decarbonylative coupling of trisubstituted cyclic olefins and benzocyclobutenones through C C activation. The metal–ligand combination [{Rh(CO)₂Cl}₂]/P(C₆F₅)₃ catalyzed this transformation most efficiently. A range of diverse spirocycles were synthesized in good to excellent yields and many sensitive functional groups were tolerated. A mechanistic study supports a hydrogen‐transfer process that occurs through a β‐H elimination/decarbonylation pathway.     

A rapid,highly efficient,and general protocol for the synthesis of functionalized triarylmethanes: a straightforward access for the synthesis of (−)-tatarinoid C

A rapid, efficient, and convenient synthesis of functionalized triarylmethane is described by the Friedel–Crafts alkylation of methoxybenzenes with a variety of aldehydes in the presence of BF₃·OEt₂. The generality of the method is demonstrated by screening a variety of di- or tri-substituted arenes as well as substituted aromatic, heteroaromatic, and aliphatic aldehydes. (−)-Tatarinoid C is synthesized in a single step following the same protocol.     

Skeletal Transformation Triggered by C−F Bond Activation after Photochemical Rearrangement of Fluorinated [7]Helicenes

Tri- and tetra-fluorinated [7]helicenes are photolabile and undergo a double fluorine atom transfer. Herein, we show that the transferred product further undergoes a skeletal transformation on silica gel. The transformation begins with activation of the allylic C−F bond on the silanol surface. Then, the resulting carbocation readily undergoes a regioselective nucleophilic aromatic substitution with water, depending on the position of the fluorine substituents. Hexafluoro-2-propanol also activated the allylic C−F bond and acted as a nucleophile. These findings support the generation of a highly reactive cationic electrophilic intermediate in the successive transformations involving fluorine atoms.     

Cobalt(II)-Catalyzed [4+2] Annulation of Picolinamides with Alkynes via C−H Bond Activation

A cobalt(II)-catalyzed [4+2] annulation of picolinamides with alkynes via C−H bond activation has been developed. The operationally simple annulation reaction allows for the synthesis of acyl-substituted 1H-benzoquinoline bearing multiple aromatic rings (up to 96 % yield) without co-oxidant or other oxidation factors under mild conditions. Several control experiments were carried out. This practical [4+2] annulation provides an efficient route to access highly functionalized compounds.     

Cobalt-Catalyzed Intramolecular Hydroacylation Involving Cyclopropane Cleavage

A simple cobalt-diphosphine catalyst has been found to efficiently promote intramolecular cyclization of ortho-cyclopropylvinyl- and cyclopropylidenemethyl-substituted benzaldehydes into benzocyclooctadienone and benzocycloheptadienone derivatives, respectively. This ring-opening hydroacylation likely involves aldehyde C−H oxidative addition, olefin insertion, cyclopropane cleavage by β-carbon elimination, and C−C bond-forming reductive elimination, as was supported by mechanistic experiments and DFT calculations.     

Minimal kernels of Dirac operators along maps

Let M be a closed spin manifold and let N be a closed manifold. For maps and Riemannian metrics g on M and h on N, we consider the Dirac operator of the twisted Dirac bundle . To this Dirac operator one can associate an index in . If M is 2‐dimensional, one gets a lower bound for the dimension of the kernel of out of this index. We investigate the question whether this lower bound is obtained for generic tupels .     

Keller–Osserman estimates and removability result for the anisotropic porous medium equation with gradient absorption term

We study “large” nonnegative solutions for a class of quasilinear equations model of which is We give a sufficient condition on the exponents and for the removability of isolated singularities.     

Explicit uniform bounds on integrals of Bessel functions and trace theorems for Fourier transforms

Explicit and partly sharp estimates are given of integrals over the square of Bessel functions with an integrable weight which can be singular at the origin. They are uniform with respect to the order of the Bessel functions and provide explicit bounds for some smoothing estimates as well as for the L² restrictions of Fourier transforms onto spheres in which are independent of the radius of the sphere. For more special weights these restrictions are shown to be Hölder continuous with a Hölder constant having this independence as well. To illustrate the use of these results a uniform resolvent estimate of the free Dirac operator with mass in dimensions is derived.     

The prescribed Ricci curvature problem on three‐dimensional unimodular Lie groups

Let G be a three‐dimensional unimodular Lie group, and let T be a left‐invariant symmetric (0,2)‐tensor field on G. We provide the necessary and sufficient conditions on T for the existence of a pair consisting of a left‐invariant Riemannian metric g and a positive constant c such that , where is the Ricci curvature of g. We also discuss the uniqueness of such pairs and show that, in most cases, there exists at most one positive constant c such that is solvable for some left‐invariant Riemannian metric g.