Rough solutions of the Einstein vacuum equationsSolutions peu régulières des équations d'Einstein dans le vide

We announce a new result, contained in [10–12], concerning rough solutions to Einstein vacuum equations expressed relative to wave coordinates. Our methods blend paradifferential techniques with a geometric approach to Strichartz type inequlities, which takes advantage of the specific structure of the Einstein equations. This results in a gain of half a derivative relative to the classical result of [3] and [5]. To cite this article: S. Klainerman, I. Rodnianski, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 125–130     

Measure functions for frames

This paper addresses the natural question: “How should frames be compared?” We answer this question by quantifying the overcompleteness of all frames with the same index set. We introduce the concept of a frame measure function: a function which maps each frame to a continuous function. The comparison of these functions induces an equivalence and partial order that allows for a meaningful comparison of frames indexed by the same set. We define the ultrafilter measure function, an explicit frame measure function that we show is contained both algebraically and topologically inside all frame measure functions. We explore additional properties of frame measure functions, showing that they are additive on a large class of supersets—those that come from so called non-expansive frames. We apply our results to the Gabor setting, computing the frame measure function of Gabor frames and establishing a new result about supersets of Gabor frames.     

Existence and asymptotic results for a system of integro-partial differential equations

A non-Fourier phase field model is considered. A global existence result for a Dirichlet, or generalized Neumann, initial-boundary value problem is obtained, followed by a discussion of the regularity and asymptotic properties of solutions ast.This research was supported in part by the National Science Foundation under Grant DMS 91-11794 and in part by the Italian M.U.R.S.T. project Problemi non lineari...Part of this author's work was done while visiting Ohio University.     

Host-guest chemistry of the chromium-wheel complex [Cr8F8(tBuCO2)16]: prediction of inclusion capabilities by using an electrostatic potential distribution determined by modeling synchrotron X-ray structure factors at 16 K

The structure and detailed electron density distribution (EDD) of the large octanuclear chromium-wheel host complex [Cr8F8(tBuCO2)16] (1) has been determined from synchrotron X-ray structure factors collected at 16(5) K. The complex has a central cavity with a minimum entry distance between carbon atoms of the pivalate methyl groups (pivalic acid = tBuCO2H) of 4.027(4) A on one side of the molecule and 7.273(4) A on the other. The screened side of the molecule can be "opened" by rotation of methyl groups to create a strained host structure, which is compensated for by improved host-guest and host-solvent interaction. The EDD of the 272-atom complex (1144 e-) was determined by multipole modeling based on the experimental structure factors. 3d orbital populations on the Cr atoms and topological analysis of the EDD show that the covalent part of the metal-ligand interactions consists mainly of sigma donation from the ligands, but that overall the interactions are predominantly electrostatic. The electrostatic potential (EP) has been calculated from the experimental EDD. Knowledge of the geometry of the naked complex 1 as well as the EP in the central cavity of this molecule allows us to deduce which characteristic properties guest molecules must have to be accepted into the void. To probe these predictions, a series of complexes of 1 with different guest inclusions were synthesized (2 = 1 + N,N'-dimethylformamide (DMF), 3 = 1 + N,N'-dimethylacetamide (DMA), 4 = 1 + DMA + DMF, 5 = 1 + 2CH3CN), and their structures were examined by using X-ray diffraction data measured at 120(1) K. Results of these studies indicate that in the crystalline state, the optimal guest molecule should be linear and possess a permanent dipole. Attempts to crystallize the host complex with cations incorporated into the cavity were fruitless, although electrospray ionization mass spectrometry showed that a [1 + potassium]+ entity pre-exists in solution and can be transferred intact into the gas phase.     

Enantiopure, supramolecular helices containing three-dimensional tetranuclear lanthanide(III) arrays: synthesis, structure, properties, and solvent-driven trinuclear/tetranuclear interconversion

The enantiomerically pure pinene-bipyridine-based receptor, (-) or (+) L(-), diastereoselectively self-assembles in dry acetonitrile in the presence of Ln(III) ions (Ln = La, Pr, Nd, Sm, Eu, Gd, and Tb) to give a C3-symmetrical, pyramidal architecture with the general formula [Ln4(L)9(mu3-OH)](ClO4)2) (abbreviated as tetra-Ln4L9). Three metal centers shape the base: an equilateral triangle surrounded by two sets of helically wrapping ligands with opposite configurations. This part of the structure is very similar to the species [Ln3(L)6(mu3-OH)(H2O)3](ClO4)2) (recently reported by us and abbreviated as tris-LnL2) formed by the ligand and the Ln(III) ions when the reactions are performed in methanol. The tetranuclear structure is completed by a capping, helical unit LnL3 whose chirality is also predetermined by the chirality of the ligand. A complete characterization of these isostructural, chiral compounds was performed in solid state (X-ray, IR) and in solution (ES-MS, NMR, CD, UV-vis and emission spectroscopies). The sign and the intensity of the CD bands in the region of the pi pi* transitions of the bipyridine (absolute Delta epsilon values at 327 nm are about 280 M(-1) x cm(-1)) are highly influenced by the helicity of the capping fragment LnL3. The photophysical properties (lifetime, quantum yield) of the visible (Eu and Tb complexes) and NIR (Nd complex) emitters indicate a good energy transfer between the ligands and the metal centers. The two related superstructures tetra-Ln4L9 and tris-LnL2 can be interconverted in acetonitrile, the switching process depending on the amount of water present in the solvent, the size of the Ln(III) ion, and the concentration. The weak chiral recognition capabilities of the self-assembly leading to the formation of tetra-Ln4L9 either by direct synthesis from a racemic mixture of the ligand and Ln(III) ions or by the conversion of a tris-Ln[(+/-)-L]2 racemate were likewise demonstrated.     

Synthesis and reactivity of the aquation product of the antitumor complex trans-[Ru(III)Cl4(indazole)2]-

Aquation of the investigational anticancer drug trans-[Ru(III)Cl4(Hind)2](-) (1, KP1019) results in the formation of mer,trans-[Ru(III)Cl3(Hind)2(H2O)] (2), which was isolated in high yield (85%) and characterized by spectroscopic methods and X-ray crystallography. Dissolution of 2 in acetone, led to its dimerization into [Ru(III)2(mu-Cl)2Cl4(Hind)4] x 2 (Me)2CO (3) in 79% yield, with release of two water molecules. Complex 2 reacts readily with nucleophilic organic molecules, viz., methanol or dimethyl sulfide, at room temperature by replacement of the aqua ligand to give mer,trans-[Ru(III)Cl3(Hind)2(MeOH)] (4) and mer,trans-[Ru(III)Cl3(Hind)2(Me2S)] (5) in 58 and 64% yield, respectively. By reaction of 2 with DMSO at room temperature or dimethyl sulfide at elevated temperatures trans,trans,trans-[Ru(II)Cl2(Hind)2(Me2S)2] (6) and trans,trans,trans-[Ru(II)Cl2(Hind)2(S-DMSO)2] (7) were prepared in 64 and 75% yield, respectively. Dissolution of 2 in acetonitrile or benzonitrile gave rise to mer,trans-[Ru(III)Cl3(Hind)(HNC(Me)ind)] (8a), mer,trans-[Ru(III)Cl3(Hind)(HNC(Ph)ind)] (8b), and trans,trans-[Ru(III)Cl2(HNC(Me)ind)2]Cl (9) in 67, 50, and 23% yield, respectively, upon metal-assisted iminoacylation of indazole, which is unprecedented for ruthenium(III). Furthermore, complex 2 reacts with the DNA-model bases 9-methyladenine (9-meade) and N6,N6-dimethyladenine (6-me2ade) to yield mer,trans-[Ru(III)Cl3(Hind)2(9-meade)] (10) and mer,trans-[Ru(III)Cl3(Hind)2(6-me2ade)] (11) with the purine bases bound to the Ru(III) center via N7 and N3, respectively. Complex 11 represents the first ruthenium complex in which the coordination of the purine ligand N6,N6-dimethyladenine occurs via N3. In addition, the polymer [Na(EtOAc)2Ru(III)(mu-Cl)4(Hind)2]n (12) was crystallized from ethyl acetate/diethyl ether solutions of Na[trans-Ru(III)Cl4(Hind)2] x 1.5 H2O (1a). The reported complexes were characterized by elemental analysis, IR and UV-vis spectroscopy, ESI mass spectrometry, cyclic voltammetry, and X-ray crystallography. Electrochemical investigations give insight into the mechanistic details of the solvolytic behavior of complex 2. The lability of the aqua ligand in 2 suggests that this complex is a potential active species responsible for the high antitumor activity of trans-[Ru(III)Cl4(Hind)2](-).     

Identification of an additional low-lying excited state of carotenoid radical cations

Carotenoids are the crucial pigments involved in photoprotection and in scavenging harmful free radicals in all living organisms. The underlying chemical processes are charge transfer and free radical reactions, both of them leading to carotenoid radical cation (Car*+) formation. Accurate knowledge of the molecular properties of Car*+ is thus a prerequisite for understanding of their function as photoprotective and antioxidant agents. Despite their fundamental importance in nonphotochemical quenching in green plants, only little is known about the Car*+ excited states and their dynamics. Our combined experimental and theoretical investigation employing femtosecond time-resolved pump-probe spectroscopy and quantum chemical calculations proves the existence of a second low-lying pipi* excited-state energetically below the well-known strongly allowed excited-state responsible for the intense absorption of Car*+ in the near-IR region. Hence, we suggest denoting the latter state as D3 state in the future. Our findings have also implications for nonphotochemical quenching in green plants, since direct quenching of chlorophyll excited states by Forster energy transfer to Car*+ is possible and efficient.     

Carotenoid radical cations as a probe for the molecular mechanism of nonphotochemical quenching in oxygenic photosynthesis

Nonphotochemical quenching (NPQ) is a fundamental mechanism in photosynthesis which protects plants against excess excitation energy and is of crucial importance for their survival and fitness. Recently, carotenoid radical cation (Car*+) formation has been discovered to be a key step for the feedback deexcitation quenching mechanism (qE), a component of NPQ, of which the molecular mechanism and location is still unknown. We have generated and characterized carotenoid radical cations by means of resonant two color, two photon ionization (R2C2PI) spectroscopy. The Car*+ bands have maxima located at 830 nm (violaxanthin), 880 nm (lutein), 900 nm (zeaxanthin), and 920 nm (beta-carotene). The positions of these maxima depend strongly on solution conditions, the number of conjugated C=C bonds, and molecular structure. Furthermore, R2C2PI measurements on the light-harvesting complex of photosystem II (LHC II) samples with or without zeaxanthin (Zea) reveal the violaxanthin (Vio) radical cation (Vio*+) band at 909 nm and the Zea*+ band at 983 nm. The replacement of Vio by Zea in the light-harvesting complex II (LHC II) has no influence on the Chl excitation lifetime, and by exciting the Chls lowest excited state, no additional rise and decay corresponding to the Car*+ signal observed previously during qE was detected in the spectral range investigated (800-1050 nm). On the basis of our findings, the mechanism of qE involving the simple replacement of Vio with Zea in LHC II needs to be reconsidered.     

Interaction of pyromellitic diimide derivatives with beta-cyclodextrin and anthracene-appended beta-cyclodextrin: rim binding vs inclusion complexation

Complex formation of pyromellitic diimide derivatives with beta-cyclodextrin and anthracene-appended beta-cyclodextrin was studied with use of induced circular dichroism and 1H NMR spectroscopies. It is revealed that pyromellitic diimides form rim-binding type complexes with beta-CD and in these complexes the pyromellitic diimides lie just outside of the narrow rim of the CD. With anthracene-appended beta-CD the pyromellitic diimides form true inclusion complexes. Implications of the formation of rim-binding type complexes are also discussed.