Graph complexes in deformation quantization

Kontsevich's formality theorem and the consequent star-product formula rely on the construction of an $L_\infty$-morphism between the DGLA of polyvector fields and the DGLA of polydifferential operators. This construction uses a version of graphical calculus. In this article we present the details of this graphical calculus with emphasis on its algebraic features. It is a morphism of differential graded Lie algebras between the Kontsevich DGLA of admissible graphs and the Chevalley-Eilenberg DGLA of linear homomorphisms between polyvector fields and polydifferential operators. Kontsevich's proof of the formality morphism is reexamined in this light and an algebraic framework for discussing the tree-level reduction of Kontsevich's star-product is described.Comment: 39 pages; 3 eps figures; uses Xy-pic. Final version. Details added, mainly concerning the tree-level approximation. Typos corrected. An abridged version will appear in Lett. Math. Phy

Understanding proof practices of pre-service mathematics teachers in geometry

In this work, we show the results of a research with pre-service mathematics secondary teachers about their Van Hiele level regarding the proof in Geometry. We observe three different profiles whose characteristics are described. These descriptions allow us to foresee certain differences when carrying out proof teaching in secondary school. The presence of a profile with a lower level than that assumed for some high school students stands out. The other two profiles show differences regarding the presence of some advanced proof strategies

Three-dimensional compressible stability-transition calculations using the spatial theory

The e(exp n)-method is employed with the spatial amplification theory to compute the onset of transition on a swept wing tested in transonic cryogenic flow conditions. Two separate eigenvalue formulations are used. One uses the saddle-point method and the other assumes that the amplification vector is normal to the leading edge. Comparisons of calculated results with experimental data show that both formulations give similar results and indicate that the wall temperature has a rather strong effect on the value of the n factor

Ultrarobust calibration of an optical lattice depth based on a phase shift

We report on a new method to calibrate the depth of an optical lattice. It consists in triggering the intrasite dipole mode of the cloud by a sudden phase shift. The corresponding oscillatory motion is directly related to the intraband frequencies on a large range of lattice depths. Remarkably, for a moderate displacement, a single frequency dominates this oscillation for the zeroth and first order interference pattern observed after a sufficiently long time-of-flight. The method is robust against atom-atom interactions and the exact value of the extra external confinement of the initial trapping potential.Comment: 7 pages, 6 figure

Geometric Hermite interpolation by rational curves of constant width

A constructive characterization of the support function for a rationally parameterized curve of constant width is given. In addition, a Hermite interpolation problem for such kind of curves is solved, which yields a method to determine a rational curve of constant width that passes through a set of free points with the corresponding tangent directions. Finally, the case of piecewise rational support functions is considered, which increases the design freedom. The procedure is presented in the general case of hedgehogs of constant width taking the advantage of projective hedgehogs, so that some constraints must be taken to ensure convexity of the desired curve.Funding for the other authors not affiliated with BCAM: Grant PID2021-124577NBI00 funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”. Project PID2019-104927GB-C21 funded by MCIN/AEI/10.13039/501100011033. Project UJI-B2022-19 funded by Universitat Jaume I. Project CIAICO/2021/180 funded by Generalitat Valenciana

Metal-Metal Cooperation in the Oxidation of a Flapping Platinum Butterfly by Haloforms: Experimental and Theoretical Evidence

The model 1-DFT for the butterfly complex [{Pt(C¿C*)(µ-pz)}2] (1; HC¿C* = 1-(4-(ethoxycarbonyl)phenyl)-3-methyl-1H-imidazol-2-ylidene) shows two minima in the potential energy surface of the ground state in acetone solution: the butterfly-wing-spreading molecules 1-s, (dPt-Pt ˜ 3.20 Å) and the wing-folding molecules 1-f (dPt-Pt = 3.00 Å). Both minima are very close in energy (¿G° = 1.7 kcal/mol) and are connected through a transition state, which lies only 1.9 kcal/mol above 1-s and 0.2 kcal/mol above 1-f. These very low barriers support a fast interconversion process, resembling a butterfly flapping, and the presence of both conformers in acetone solution. However, the 1-f ratio is so low that it is undetectable in the excitation and emission spectra of 1 in 2-MeTHF of diluted solutions (10-5 M) at 77 K, while it is seen in more concentrated solutions (10-3 M). In acetone solution, 1 undergoes a [2c, 2e] oxidation by CHX3 (X = Cl, Br) in the sunlight to render the Pt2(III, III) compounds [{Pt(C¿C*)(µ-pz)X)}2] (X = Cl (2-Cl), Br (2-Br)). In concentrated solutions, 1 can react with CHCl3 under blue light to give 2-Cl and with CHBr3 in the dark, the latter rendering the compound [BrPt(C¿C*)(µ-pz)2Pt(C¿C*)CHBr2] (3-Br) or mixtures of 2-Br and 3-Br if the reaction is performed under an argon atmosphere or in the air, respectively. Mechanistic studies showed that in concentrated solutions the oxidation processes follow a radical mechanism being the MMLCT-based species 1-f, those which trigger the reaction of 1 with CHBr3 and CHCl3. In the ground state (S0f), it promotes the thermal oxidation of 1 by CHBr3 and in the first singlet excited state (S1f) the blue-light-driven photooxidation of 1 by CHCl3. Complexes, 2-Cl, 2-Br, and 3-Br were selectively obtained and fully characterized, showing Pt-Pt distances (ca. 2.6 Å) shorter than that of the starting complex, 1. They are, together with the analogous [{Pt(C¿C*)(µ-pz)I)}2] and [IPt(C¿C*)(µ-pz)2Pt(C¿C*)CHI2], the only dinuclear metal-metal-bonded PtIII(µ-pz)2PtIII compounds reported to date

High-Valent Pyrazolate-Bridged Platinum Complexes: A Joint Experimental and Theoretical Study

Complexes {Pt(C^C*)(µ-pz)}2] (HC^C*A= 1-(4-(ethoxycarbonyl)phenyl)-3-methyl-1H-imidazol-2-ylidene 1a, HC^C*B= 1-phenyl-3-methyl-1H-imidazol-2-ylidene 1b) react with methyl iodide (MeI) at room temperature in the dark to give compounds {PtIV(C^C*)Me(µ-pz)}2(µ-I)]I (C^C*A2a, C^C*B2b). The reaction of 1a with benzyl bromide (BnBr) in the same conditions afforded Br(C^C*A)PtIII(µ-pz)2PtIII(C^C*A)Bn] (5a), which by heating in BnBr(l) became {PtIV(C^C*A)Bn(µ-pz)}2(µ-Br)]Br (6a). Experimental investigations and density functional theory (DFT) calculations on the mechanisms of these reactions from 1a revealed that they follow a SN2 pathway in the two steps of the double oxidative addition (OA). Based on the DFT investigations, species such as (C^C*A)PtIII(µ-pz)2PtIII(C^C*A)R]X (RX = MeI Int-Me, BnBr Int-Bn) and (C^C*A)PtII(µ-pz)2PtIV(C^C*A)(R)X] (RX = MeI Int'-Me, BnBr Int'-Bn) were proposed as intermediates for the first and the second OA reactions, respectively. In order to put the mechanisms on firmer grounds, Int-Me was prepared as (C^C*A)PtIII(µ-pz)2PtIII(C^C*A)Me]BF4(3a') and used to get I(C^C*A)PtIII(µ-pz)2PtIII(C^C*A)Me](4a), (C^C*A)PtII(µ-pz)2PtIV(C^C*A)(Me)I](Int'-Me), and {PtIV(C^C*)Me(µ-pz)}2(µ-I)]BF4(2a'). The single-crystal X-ray structures of 2a, 2b, 3a', and 5a along with the mono- and bi-dimensional 1H and 195Pt{1H} NMR spectra of all the named species allowed us to compare structural and spectroscopic data for high-valent complexes with the same core {Pt(C^C*)(µ-pz)}2] but different oxidation states. © 2022 American Chemical Society. All rights reserved

Metabolites, genomics, epigenomics, exposomics and health: Focus on serum bilirubin concentrations in subjects with metabolic syndrome from a Mediterranean population

Pòster presentat al congrés "Understanding Human Diseases Through Metabolomics: Interactions Among the Genome, Proteome, Gut Microbiome and Nutrition", Metabolomics and Human Health, Gordon Research Conference (February 3 - 8, 2019 Ventura, CA, United States)Although metabolomics aims at the measurement of small molecules (metabolites) in a biological sample, this knowledge requires additional information on the related genetic variants, epigenetic regulators and environmental factors (diet, smoking, physical activity, etc.) in order to translate the knowledge into actionable therapeutic or preventive evidence for complex disease outcomes. We focused on serum bilirubin, a metabolite generated when heme oxygenase catalyzes the degradation of heme (Figure 1). This produces biliverdin, which is converted into bilirubin by biliverdinreductase. Bilirubin is further processed in hepatocytes, where unconjugated bilirubin is conjugated by uridine diphosphate-glucuronosyltransferase (UDP-GT) to a water-soluble form for excretion. For decades, increased serum bilirubin concentrations were considered a threatening sign of underlying liver disease and had been associated with neonatal jaundice. However, data from recent years show that bilirubin is a powerful antioxidant and suggest that slightly increased serum bilirubin concentrations are protective against oxidative stress-related diseases