The geometry of the asymptotics of polynomial maps

The aim of this paper is to develop a theory for the asymptotic behavior of polynomials and of polynomial maps overR and overC and to apply it to the Jacobian conjecture. This theory gives a unified frame for some results on polynomial maps that were not related before. A well known theorem of J. Hadamard gives a necessary and sufficient condition on a local diffeomorphismf: R ⁿ →R ⁿ to be a global diffeomorphism. In order to show thatf is a global diffeomorphism it suffices to exclude the existence of asymptotic values forf. The real Jacobian conjecture was shown to be false by S. Pinchuk. Our first application is to understand his construction within the general theory of asymptotic values of polynomial maps and prove that there is no such counterexample for the Jacobian conjecture overC. In a second application we reprove a theorem of Jeffrey Lang which gives an equivalent formulation of the Jacobian conjecture in terms of Newton polygons. This generalizes a result of Abhyankar. A third application is another equivalent formulation of the Jacobian conjecture in terms of finiteness of certain polynomial rings withinC[U, V]. The theory has a geometrical aspect: we define and develop the theory of etale exotic surfaces. The simplest such surface corresponds to Pinchuk's construction in the real case. In fact, we prove one more equivalent formulation of the Jacobian conjecture using etale exotic surfaces. We consider polynomial vector fields on etale exotic surfaces and explore their properties in relation to the Jacobian conjecture. In another application we give the structure of the real variety of the asymptotic values of a polynomial mapf: R ² →R ² .     

Carbon Dioxide Reduction on Transition Metal Dichalcogenides with Ni and Cu Edge Doping: A Density-Functional Theory Study

Transition-metal dichalcogenides (TMDs) have promising properties for their use as catalysts of CO₂ reduction to methane via the Sabatier reaction. In this article we use density-functional theory calculations to gain insight into the energetics of this reaction for Mo/W-based and S/Se-based TMDs with non-, Ni- and Cu-doping. We show that sulfur-based TMDs with Ni/Cu doping exhibit better indicators for catalytic performance of the CO₂ reduction reaction than non-doped and doped TMDs without active sites. In addition, the role of the transition metal was found to a much smaller influence in the reaction than the role of the chalcogen and dopant atoms, which influence the bonding strength and type, respectively.     

Targeting large kinase active site with rigid, bulky octahedral ruthenium complexes

A strategy for targeting protein kinases with large ATP-binding sites by using bulky and rigid octahedral ruthenium complexes as structural scaffolds is presented. A highly potent and selective GSK3 and Pim1 half-sandwich complex NP309 was successfully converted into a PAK1 inhibitor by making use of the large octahedral compounds Lambda-FL172 and Lambda-FL411 in which the cyclopentadienyl moiety of NP309 is replaced by a chloride and sterically demanding diimine ligands. A 1.65 A cocrystal structure of PAK1 with Lambda-FL172 reveals how the large coordination sphere of the ruthenium complex matches the size of the active site and serves as a yardstick to discriminate between otherwise closely related binding sites.     

Radial and angular rotons in trapped dipolar gases

We study Bose-Einstein condensates with purely dipolar interactions in oblate traps. We find that the condensate always becomes unstable to collapse when the number of particles is sufficiently large. We analyze the instability, and find that it is the trapped-gas analogue of the "roton-maxon" instability previously reported for a gas that is unconfined in 2D. In addition, we find that under certain circumstances the condensate wave function attains a biconcave shape, with its maximum density away from the center of the gas. These biconcave condensates become unstable due to azimuthal excitation--an angular roton.     

Aggregation and Cooperative Effects in the Aldol Reactions of Lithium Enolates

Density functional theory and Car–Parrinello molecular dynamics simulations have been carried out for model aldol reactions involving aggregates of lithium enolates derived from acetaldehyde and acetone. Formaldehyde and acetone have been used as electrophiles. It is found that the geometries of the enolate aggregates are in general determined by the most favorable arrangements of the point charges within the respective Li_nO_n clusters. The reactivity of the enolates follows the sequence monomer?dimer>tetramer. In lithium aggregates, the initially formed aldol adducts must rearrange to form more stable structures in which the enolate and alkoxide oxygen atoms are within the respective Li_nO_n clusters. Positive cooperative effects, similar to allosteric effects found in several proteins, are found for the successive aldol reactions in aggregates. The corresponding transition structures show in general sofa geometries.     

Structurally sophisticated octahedral metal complexes as highly selective protein kinase inhibitors

The generation of synthetic compounds with exclusive target specificity is an extraordinary challenge of molecular recognition and demands novel design strategies, in particular for large and homologous protein families such as protein kinases with more than 500 members. Simple organic molecules often do not reach the necessary sophistication to fulfill this task. Here, we present six carefully tailored, stable metal-containing compounds in which unique and defined molecular geometries with natural-product-like structural complexity are constructed around octahedral ruthenium(II) or iridium(III) metal centers. Each of the six reported metal compounds displays high selectivity for an individual protein kinase, namely GSK3α, PAK1, PIM1, DAPK1, MLCK, and FLT4. Although being conventional ATP-competitive inhibitors, the combination of the unusual globular shape and rigidity characteristics, of these compounds facilitates the design of highly selective protein kinase inhibitors. Unique structural features of the octahedral coordination geometry allow novel interactions with the glycine-rich loop, which contribute significantly to binding potencies and selectivities. The sensitive correlation between metal coordination sphere and inhibition properties suggests that in this design, the metal is located at a "hot spot" within the ATP binding pocket, not too close to the hinge region where globular space is unavailable, and at the same time not too far out toward the solvent where the octahedral coordination sphere would not have a significant impact on potency and selectivity. This study thus demonstrates that inert (stable) octahedral metal complexes are sophisticated structural scaffolds for the design of highly selective chemical probes.     

On the Lifetime of the Transients (NP)?(CH3)n (NP=Ag0, Au0, TiO2 Nanoparticles) Formed in the Reactions Between Methyl Radicals and Nanoparticles Suspended in Aqueous Solutions

Methyl radicals react in fast reactions, with rate constants k>1×10⁸ M ^?1 s^?1, with Au⁰, Ag⁰ and TiO₂ nanoparticles (NPs) dispersed in aqueous solutions to form intermediates, (NP)? (CH₃)_n, in which the methyl groups are covalently bound to the NPs. These intermediates decompose to form ethane. As n≥2 is required for the formation of C₂H₆, the minimal lifetime (τ) of the methyls bound to the NPs, (NP)? CH₃, can be estimated from the rate of production of the CH₃^. radicals and the NPs concentration. The results obtained in this study, using a very low dose rate γ‐source for NP=Ag⁰, Au⁰, and TiO₂ point out that τ of these intermediates is surprisingly long, for example, ≥8 and ≥188 sec for silver and gold, respectively. These data point out that the NP? C bond dissociation energies are ≥70 kJ mol^?1. Under low rates of production of CH₃^., that is, when the rate of formation of ethane is very low, other reactions may occur, consequently the mechanism proposed is “broken”. This is observed in the present study only for TiO₂ NPs. These results have to be considered whenever alkyl radicals are formed near surfaces. Furthermore, the results point out that the rate of reaction of methyl radicals with (NP)? (CH₃)_n depends on n, that is, the number of methyl radicals bound to the NPs affect the properties of the NPs.