Anosov and expanding attractors

Science China Mathematics - An earlier conjecture is settled with an immersion of a 2-dimensional branched manifold. Possible obstructions in linear algebra and tiling theory are studied first.     

On Concavity of the Monopolist's Problem Facing Consumers with Nonlinear Price Preferences

A monopolist wishes to maximize her profits by finding an optimal price policy. After she announces a menu of products and prices, each agent x will choose to buy that product y(x) which maximizes his own utility, if positive. The principal's profits are the sum of the net earnings produced by each product sold. These are determined by the costs of production and the distribution of products sold, which in turn are based on the distribution of anonymous agents and the choices they make in response to the principal's price menu. In this paper, we provide a necessary and sufficient condition for the convexity or concavity of the principal's (bilevel) optimization problem, assuming each agent's disutility is a strictly increasing but not necessarily affine (i.e., quasilinear) function of the price paid. Concavity, when present, makes the problem more amenable to computational and theoretical analysis; it is key to obtaining uniqueness and stability results for the principal's strategy in particular. Even in the quasilinear case, our analysis goes beyond previous work by addressing convexity as well as concavity, by establishing conditions which are not only sufficient but necessary, and by requiring fewer hypotheses on the agents' preferences. © 2019 Wiley Periodicals, Inc.     

Catalytic Hydroetherification of Unactivated Alkenes Enabled by Proton-Coupled Electron Transfer

We report a catalytic, light-driven method for the intramolecular hydroetherification of unactivated alkenols to furnish cyclic ether products. These reactions occur under visible-light irradiation in the presence of an Ir^III-based photoredox catalyst, a Brønsted base catalyst, and a hydrogen-atom transfer (HAT) co-catalyst. Reactive alkoxy radicals are proposed as key intermediates, generated by direct homolytic activation of alcohol O−H bonds through a proton-coupled electron-transfer mechanism. This method exhibits a broad substrate scope and high functional-group tolerance, and it accommodates a diverse range of alkene substitution patterns. Results demonstrating the extension of this catalytic system to carboetherification reactions are also presented.     

Synthesis,characterization, and cytotoxic activity studies of new N4O complexes derived from 2-({3-[2-morpholinoethylamino]-N3-([pyridine-2-yl]methyl) propylimino} methyl)phenol

A new unsymmetrical five-coordinate Schiff base ligand (HL) with an N₄O donor set ( 2 ) has been prepared by condensation of N1-(2-morpholinoethyl)-N1-([pyridine-2-yl]methyl)propane-1,3-diamine with 2-hydroxy-benzaldehyde. Metal complexes [ML]ⁿ⁺ (M = Zn²⁺, Cd²⁺, Mn²⁺, Cu²⁺, Ni²⁺, Ag⁺, Fe³⁺, and Co²⁺ ( 3–10 ) were synthesized by the reaction of the ligand and metal salts in ethanol. The resulting products were characterized by elemental analyses, infrared, ¹H and ¹³C nuclear magnetic resonance spectra (in the case of Cd and Zn complexes), UV–Vis, electrospray ionization-mass spectrometric, and conductivity measurements. The structure of the complexes [ZnL](ClO₄) ( 3 ), [CdL](ClO₄) ( 4 ), and [CuL](ClO₄) ( 7 ) has been determined by single-crystal X-ray diffraction analysis. The metal complexes were determined to have a distorted trigonal bipyramidal (Zn and Cd) or a distorted square pyramidal (Cu) geometry. The cytotoxic potential of each compound (1–10) against MCF-7 and MDA-MB-231 (breast cancer cells), PC-3 (prostate cancer cells), and WI-38 human normal lung fibroblast cells was evaluated using the MTT (3-[4,5-dimethylthiazol-2-yl]-2,5-diphenyl tetrazolium bromide) assay. Compounds 1, 2, and 10 did not display any activity toward any cell line tested. None of the compounds except compound 8 was cytotoxic toward PC-3. Compounds 4 and 8 showed the highest cytotoxic activity against the MCF-7 and MDA-MB-231 cell lines. Because compounds 3, 6, and 9 have similar half-maximal inhibitory concentration values against cancer cells and normal cells, these compounds displayed poor selectivity between cancer and normal cells. More importantly, it was observed that compound 5 acts differently toward different types of cell lines. For example, it displays lower cytotoxicity against the WI-38 normal cell line than it does against the MDA-MB-231 cell line.     

Synthesis,Structure, and DFT Analysis of the THF Solvate of 2-Picolyllithium: A 2-Picolyllithium Solvate with Significant Carbanionic Character

Previous studies of different solvates of 2-methylpyridyllithium (2-picolyllithium) have uncovered electronic structures corresponding to aza-allyl and enamido resonance forms of the metallated pyridine-based compounds. Here, we report the synthesis and characterization of [2-CH₂Li(THF)₂C₅H₄N], a new THF solvate. X-ray crystallographic studies reveal a dimeric arrangement featuring a non-planar eight-membered [NCCLi]₂ ring, in which the primary cation-anion interaction is between the central Li atom and the C atom of the deprotonated methyl group [length, 2.285(2) Å], suggesting a new carbanionic resonance structure for this 2-picolyllithium series. The significant carbanionic character of [2-CH₂Li(THF)₂C₅H₄N] was confirmed by gas-phase DFT calculations [B3LYP/6-311+G(d)] with the calculated electron density interrogated by means of quantum theory of atoms in molecules (QTAIM) and natural bond orbital (NBO) analyses. For comparison these computational analyses were also performed on the literature structures of [2-CH₂Li(2-Picoline)C₅H₄N] and [2-CH₂Li(PMDETA)C₅H₄N]. In a reactivity study, [2-CH₂Li(THF)₂C₅H₄N] was found to undergo nucleophilic addition to pyridine to generate dipyridylmethane in a good yield.     

3-O-Sulfation of Heparan Sulfate Enhances Tau Interaction and Cellular Uptake

Prion-like transcellular spreading of tau in Alzheimer's Disease (AD) is mediated by tau binding to cell surface heparan sulfate (HS). However, the structural determinants for tau–HS interaction are not well understood. Microarray and SPR assays of structurally defined HS oligosaccharides show that a rare 3-O-sulfation (3-O-S) of HS significantly enhances tau binding. In Hs3st1^−/− (HS 3-O-sulfotransferase-1 knockout) cells, reduced 3-O-S levels of HS diminished both cell surface binding and internalization of tau. In a cell culture, the addition of a 3-O-S HS 12-mer reduced both tau cell surface binding and cellular uptake. NMR titrations mapped 3-O-S binding sites to the microtubule binding repeat 2 (R2) and proline-rich region 2 (PRR2) of tau. Tau is only the seventh protein currently known to recognize HS 3-O-sulfation. Our work demonstrates that this rare 3-O-sulfation enhances tau–HS binding and likely the transcellular spread of tau, providing a novel target for disease-modifying treatment of AD and other tauopathies.