Characterization of ellipses as uniformly dense sets with respect to a family of convex bodies

Let \(K\subset \mathbb R ^N\) be a convex body containing the origin. A measurable set \(G\subset \mathbb R ^N\) with positive Lebesgue measure is said to be uniformly \(K\) -dense if, for any fixed \(r>0\) , the measure of \(G\cap (x+r K)\) is constant when \(x\) varies on the boundary of \(G\) (here, \(x+r K\) denotes a translation of a dilation of \(K\) ). We first prove that \(G\) must always be strictly convex and at least \(C^{1,1}\) -regular; also, if \(K\) is centrally symmetric, \(K\) must be strictly convex, \(C^{1,1}\) -regular and such that \(K=G-G\) up to homotheties; this implies in turn that \(G\) must be \(C^{2,1}\) -regular. Then for \(N=2\) , we prove that \(G\) is uniformly \(K\) -dense if and only if \(K\) and \(G\) are homothetic to the same ellipse. This result was already proven by Amar et al. in 2008 . However, our proof removes their regularity assumptions on \(K\) and \(G\) , and more importantly, it is susceptible to be generalized to higher dimension since, by the use of Minkowski’s inequality and an affine inequality, avoids the delicate computations of the higher-order terms in the Taylor expansion near \(r=0\) for the measure of \(G\cap (x+r\,K)\) (needed in 2008).     

Towards analytically useful two-dimensional Fourier transform ion cyclotron resonance mass spectrometry

Fourier transform ion cyclotron resonance (FT-ICR) mass spectrometry (MS) achieves high resolution and mass accuracy, allowing the identification of the raw chemical formulae of ions in complex samples. Using ion isolation and fragmentation (MS/MS), we can obtain more structural information, but MS/MS is time- and sample-consuming because each ion must be isolated before fragmentation. In 1987, Pfändler et al. proposed an experiment for 2D FT-ICR MS in order to fragment ions without isolating them and to visualize the fragmentations of complex samples in a single 2D mass spectrum, like 2D NMR spectroscopy. Because of limitations of electronics and computers, few studies have been conducted with this technique. The improvement of modern computers and the use of digital electronics for FT-ICR hardware now make it possible to acquire 2D mass spectra over a broad mass range. The original experiments used in-cell collision-induced dissociation, which caused a loss of resolution. Gas-free fragmentation modes such as infrared multiphoton dissociation and electron capture dissociation allow one to measure high-resolution 2D mass spectra. Consequently, there is renewed interest to develop 2D FT-ICR MS into an efficient analytical method. Improvements introduced in 2D NMR spectroscopy can also be transposed to 2D FT-ICR MS. We describe the history of 2D FT-ICR MS, introduce recent improvements, and present analytical applications to map the fragmentation of peptides. Finally, we provide a glossary which defines a few keywords for the 2D FT-ICR MS field.     

Palladium-Catalyzed Cross-Coupling of Gem-Bromofluoroalkenes with Alkylboronic Acids for the Synthesis of Alkylated Monofluoroalkenes

Monofluoroalkenes are versatile fluorinated synthons in organic synthesis, medicinal chemistry and materials science. In light of the importance of alkyl-substituted monofluoroalkenes efficient synthesis of these moieties still represents a synthetic challenge. Herein, we described a mild and efficient methodology to obtain monofluoroalkenes through a stereospecific palladium-catalyzed alkylation of gem-bromofluoroalkenes with primary and strained secondary alkylboronic acids under mild conditions. This novel strategy gives access to a wide range of functionalized tri- and tetrasubstituted monofluoroalkenes in high yield, with good functional group tolerance, independently from the gem-bromofluoroalkenes geometry.     

C(sp3)−H Oxidative Addition and Transfer Hydrogenation Chemistry of a Titanium(II) Synthon: Mimicry of Late‐Metal Type Reactivity

Two‐electron reduction of the Ti^IV compound (^ketguan)(Im^DippN)Ti(OTf)₂ ( 3 ) gives the arene‐masked complex (^ketguan)(η⁶‐Im^DippN)Ti ( 1 ) in excellent yield. Upon standing in solution, 1 converts to a Ti^IV metallacycle ( 4 ) through dehydrogenation of a pendant isopropyl group. Spectroscopic evidence shows this transformation initially proceeds via the oxidative addition of a C(sp³)?H bond and can be reversed upon exposure of 4 to H₂. Interestingly, treatment of 1 with cyclohexene gives cyclohexane and 4 via a titanium‐mediated transfer hydrogenation reaction, a process that can be extended to catalytically hydrogenate other unsaturated hydrocarbons under mild conditions. These results, rare for the early‐metals, suggest 1 possesses chemical characteristics reminiscent of noble, late‐metals.     

Incorporation of the HERG potassium channel in a mercury supported lipid bilayer

The HERG potassium channel was incorporated in a mercury-supported tethered bilayer lipid membrane (tBLM) obtained by anchoring a thiolipid monolayer to the mercury surface and by self-assembling a lipid monolayer on top of it from a lipid film spread on the surface of an electrolyte solution. HERG was then incorporated in this tBLM from its micellar solution in Triton X-100, thus avoiding the use of vesicles in the preparation of the tBLM and of proteoliposomes in channel incorporation. The HERG "inward" current following a repolarization step was obtained by subtracting the current recorded upon addition of the specific inhibitor WAY from that recorded prior to this addition. This current was compared with that reported in the literature by the patch-clamp technique.     

Pi-stacking effect on levoglucosenone derived internal chiral auxiliaries. A case of complete enantioselectivity inversion on the Diels-Alder reaction

Detailed quantum chemical calculations, experimental evidence, and NMR data rationalize the participation of pi-stacking interaction in the highly asymmetric Diels-Alder reaction using levoglucosenone derived internal chiral auxiliaries, including the appealing effect of inversion of the enantioselectivity by coordination of the substrate with Et 2AlCl.     

A proximal method with logarithmic barrier for nonlinear complementarity problems

We study the proximal method with the regularized logarithmic barrier, originally stated by Attouch and Teboulle for positively constrained optimization problems, in the more general context of nonlinear complementarity problems with monotone operators. We consider two sequences generated by the method. We prove that one of them, called the ergodic sequence, is globally convergent to the solution set of the problem, assuming just monotonicity of the operator and existence of solutions; for convergence of the other one, called the proximal sequence, we demand some stronger property, like paramonotonicity of the operator or the so called “cut property” of the problem.     

Compactness and existence results in weighted Sobolev spaces of radial functions. Part II: existence

We apply the compactness results obtained in the first part of this work, to prove existence and multiplicity results for finite energy solutions to the nonlinear elliptic equation

$$-\triangle u + V \left(\left|x\right|\right) u = g \left(\left|x\right|, u\right) \quad {\rm in} \Omega \subseteq \mathbb{R}^{N},\,N \geq 3,$$

where \({\Omega}\) is a radial domain (bounded or unbounded) and u satisfies u =  0 on \({\partial\Omega}\) if \({\Omega \neq\mathbb{R}^{N}}\) and \({u \rightarrow 0}\) as \({\left|x\right| \rightarrow \infty}\) if \({\Omega}\) is unbounded. The potential V may be vanishing or unbounded at zero or at infinity and the nonlinearity g may be superlinear or sublinear. If g is sublinear, the case with a forcing term \({g\left(\left|\cdot\right|, 0\right) \neq 0}\) is also considered. Our results allow to deal with V and g exhibiting behaviours at zero or at infinity which are new in the literature and, when \({\Omega = \mathbb{R}^{N}}\), do not need to be compatible with each other.

     

W*-rigidity for the von Neumann algebras of products of hyperbolic groups

We show that if \({\Gamma = \Gamma_1\times\dotsb\times \Gamma_n}\) is a product of \({{\rm n} \geq 2}\) non-elementary ICC hyperbolic groups then any discrete group \({\Lambda}\) which is \({W^*}\)-equivalent to \({\Gamma}\) decomposes as a direct product of n ICC groups and does not decompose as a direct product of k ICC groups when \({{\rm n} \not= {\rm k}}\). This gives a group-level strengthening of Ozawa and Popa’s unique prime decomposition theorem by removing all assumptions on the group \({\Lambda}\). This result in combination with Margulis’ normal subgroup theorem allows us to give examples of lattices in the same Lie group which do not generate stably equivalent II₁ factors.