Mehler's formula and functional calculus Dedicated to Professor Jean-Yves Chemin on the Occasion of His 60th Birthday

We show that Mehler's formula can be used to handle several formulas involving the quantization of singular Hamiltonians. In particular, we diagonalize in the Hermite basis the Weyl quantization of the characteristic function of several domains of the phase space.     

Synthetically Tuning the 2‐Position of Halogenated Quinolines: Optimizing Antibacterial and Biofilm Eradication Activities via Alkylation and Reductive Amination Pathways

Agents capable of eradicating bacterial biofilms are of great importance to human health as biofilm‐associated infections are tolerant to our current antibiotic therapies. We have recently discovered that halogenated quinoline (HQ) small molecules are: 1) capable of eradicating methicillin‐resistant Staphylococcus aureus (MRSA), methicillin‐resistant Staphylococcus epidermidis (MRSE) and vancomycin‐resistant Enterococcus faecium (VRE) biofilms, and 2) synthetic tuning of the 2‐position of the HQ scaffold has a significant impact on antibacterial and antibiofilm activities. Here, we report the chemical synthesis and biological evaluation of 39 HQ analogues that have a high degree of structural diversity at the 2‐position. We identified diverse analogues that are alkylated and aminated at the 2‐position of the HQ scaffold and demonstrate potent antibacterial (MIC≤0.39 μm ) and biofilm eradication (MBEC 1.0–93.8 μm ) activities against drug‐resistant Staphylococcus aureus, Staphylococcus epidermidis and Enterococcus faecium strains while demonstrating <5 % haemolysis activity against human red blood cells (RBCs) at 200 μm . In addition, these HQs demonstrated low cytotoxicity against HeLa cells. Halogenated quinolines are a promising class of antibiofilm agents against Gram‐positive pathogens that could lead to useful treatments against persistent bacterial infections.     

Four (2,2′‐bipyridyl)(ferrocenyl)boronium derivatives

Crystal structures are reported for four (2,2′‐bipyridyl)(ferrocenyl)boronium derivatives, namely (2,2′‐bipyridyl)(ethenyl)(ferrocenyl)boronium hexafluoridophosphate, [Fe(C₅H₅)(C₁₇H₁₅BN₂)]PF₆, (Ib), (2,2′‐bipyridyl)(tert‐butylamino)(ferrocenyl)boronium bromide, [Fe(C₅H₅)(C₁₉H₂₂BN₃)]Br, (IIa), (2,2′‐bipyridyl)(ferrocenyl)(4‐methoxyphenylamino)boronium hexafluoridophosphate acetonitrile hemisolvate, [Fe(C₅H₅)(C₂₂H₂₀BN₃O)]PF₆·0.5CH₃CN, (IIIb), and 1,1′‐bis[(2,2′‐bipyridyl)(cyanomethyl)boronium]ferrocene bis(hexafluoridophosphate), [Fe(C₁₇H₁₄BN₃)₂](PF₆)₂, (IVb). The asymmetric unit of (IIIb) contains two independent cations with very similar conformations. The B atom has a distorted tetrahedral coordination in all four structures. The cyclopentadienyl rings of (Ib), (IIa) and (IIIb) are approximately eclipsed, while a bisecting conformation is found for (IVb). The N—H groups of (IIa) and (IIIb) are shielded by the ferrocenyl and tert‐butyl or phenyl groups and are therefore not involved in hydrogen bonding. The B—N(amine) bond lengths are shortened by delocalization of π‐electrons. In the cations with an amine substituent at boron, the B—N(bipyridyl) bonds are 0.035 (3) Å longer than in the cations with a methylene C atom bonded to boron. A similar lengthening of the B—N(bipyridyl) bonds is found in a survey of related cations with an oxy group attached to the B atom.     

Fast rotating Bose-Einstein condensates in an asymmetric trap

We investigate the effect of the anisotropy of a harmonic trap on the behaviour of a fast rotating Bose-Einstein condensate. This is done in the framework of the 2D Gross-Pitaevskii equation and requires a symplectic reduction of the quadratic form defining the energy. This reduction allows us to simplify the energy on a Bargmann space and study the asymptotics of large rotational velocity. We characterize two regimes of velocity and anisotropy; in the first one where the behaviour is similar to the isotropic case, we construct an upper bound: a hexagonal Abrikosov lattice of vortices, with an inverted parabola profile. The second regime deals with very large velocities, a case in which we prove that the ground state does not display vortices in the bulk, with a 1D limiting problem. In that case, we show that the coarse grained atomic density behaves like an inverted parabola with large radius in the deconfined direction but keeps a fixed profile given by a Gaussian in the other direction. The features of this second regime appear as new phenomena.     

A note on weighted bounds for rough singular integrals

We show that the $L^2(w)$ operator norm of the composition $M°T_{\mathrm{\Omega }}$, where M is the maximal operator and $T_{\mathrm{\Omega }}$ is a rough homogeneous singular integral with angular part $\mathrm{\Omega }\in L^{\infty }(S^{n?1})$, depends quadratically on ${[w]}_{A_2}$, and that this dependence is sharp.     

The twinned crystal structure of tripotassium benzene‐1,3,5‐tris(trifluoroborate)

The title compound, 3K⁺·C₆H₃B₃F₉³⁻, crystallizes as discrete anions and cations which are connected by K...F and K...π interactions. Two of the –BF₃ residues attached to the aromatic ring adopt a conformation with all F atoms out of the plane of the aromatic ring, whereas the third residue has an almost synperiplanar conformation for one of the F—B—C—C torsion angles. It is remarkable that only one of the K⁺ cations interacts with the arene ring and that only one side of the aromatic ring coordinates to a K⁺ cation. As a result, a sandwich structure does not occur. All K⁺ ions show a coordination mode that cannot be conveniently described with a polyhedron. The anions are located in the (102) planes with the K⁺ cations located between these planes. The investigated crystal was a nonmerohedral twin with the fractional contribution of the minor twin component being 0.405 (4). The title compound is the first example of a structure containing a benzene ring substituted with three –BF₃ groups. Only eight other structures have been reported in which a benzene ring carries at least one –BF₃ group. Just five of these contain a K⁺ ion, but in none of these is the K⁺ ion coordinated to the aromatic ring.