Subdifferentials of compactly lipschitzian vector-valued functions

Summary We introduce the concept of compactly lipschitzian functions taking values in a topological vector space F. We show that if F is finite dimensional the Lipschitz functions are compactly lipschitizian. We define the notions of generalized directional derivatives and subdifferentials for such functionsf taking values in an ordered topological vector space. It is shown that this notion of subdifferential coincides with the one of F. H. Clarke whenf is Lispchits and F=. Some formulas for this subdifferential concerning the cases of finite sum, composition, pointwise supremum and continuous sum are studied.     

Tris[(1‐methylimidazol‐2‐yl)methyl]amine–boric acid (1/1)

Cocrystallization of a poly­imidazole compound with boric acid results in the formation of the title compound, C₁₅H₂₁N₇·B(OH)₃, which has an extensive hydrogen‐bonding network. The O?N(im) separations (im is imidazole) range from 2.6991 (15) to 2.7914 (14) Å, with O—H?N angles ranging from 170.6 (18) to 175 (2)°. In addition, symmetry‐related boric acid mol­ecules form intermolecular hydrogen bonds, with an O?O distance of 2.7582 (14) Å, and symmetry‐related imidazole groups form π–π stacks, with a centroid‐to‐centroid separation of 3.533 Å.     

Ab initio simulation of benzene Raman intensities

C₆H₆ Raman scattering activities calculated from harmonic model ab initio Hartree–Fock 6–311 ++ G(d, p) polarizability derivatives (and harmonic force fields calculated at the same level) accurately simulate experiment (to within 1% for the a_1g modes). Accurate predictions are also made for the e_2g modes (to within 5% for ν₇ and ν₉, and more poorly for ν₆ and ν₈ [in Fermi resonance with ν₆ + ν₁]) and for the e_1g out-of-plane mode, ν₁₀. Only the ν₆ in-plane CCC bending mode scattering activity is found to be anomalous. Systematic variation of the basis set indicates that the benzene scattering activities and depolarization ratios are strongly dependent on inclusion of both carbon and hydrogen atom diffuse functions in the basis set. Predictions are also made for ¹²C₆D₆ and for unmeasured intensities in ¹³C₆H₆. Measurements of a_1g mode scattering activities in the latter molecule are predicted to be useful in testing the harmonic Hartree–Fock Raman intensity model.     

Trinuclear Schiff base complexes with uranium(V) and copper(II) or zinc(II) ions

Treatment of the uranium(IV) complexes [{ML¹(py)}₂U^IV] (M = Cu, Zn; L¹ = N,N′-bis(3-hydroxysalicylidene)-1,3-propanediamine) with silver nitrate in pyridine led to the formation of the corresponding cationic uranium(V) species which were found to be thermally unstable and were converted back into the parent U^IV complexes; no electron transfer was observed in solution between the U^IV and U^V compounds. In the crystals of [{ML¹(py)}₂U^IV][{ML¹(py)}₂U^V][NO₃], the neutral U^IV and cationic U^V species are clearly identified by the distinct U–O distances. Similar reaction of [{ZnL²(py)}₂U^IV] [L² = N,N′-bis(3-hydroxysalicylidene)-1,4-butanediamine] with AgNO₃ gave crystals of [{ZnL²(py)}U^V{ZnL²(py)₂}][NO₃] but the copper counterpart was not isolated. Crystals of [{ZnL¹(py)}₂U^V][OTf] · THF (OTf = OSO₂CF₃) were obtained fortuitously from the reaction of [Zn(H₂L¹)] and U(OTf)₃.     

Multi-State Second-Order Nonlinear Optical Switches Incorporating One to Three Benzazolo-Oxazolidine Units: A Quantum Chemistry Investigation

This contribution employs quantum chemistry methods to describe the variations of the second nonlinear optical responses of molecular switches based on benzazolo-oxazolidine (BOX) units, connected by π-linkers, along their successive opening/closing. Under the fully closed forms, all of them display negligible first hyperpolarizability (β) values. When one BOX is opened, which is sketched as C→O, a push–pull π-conjugated segment is formed, having the potential to enhance β and to set the depolarization ratio (DR) to its one-dimensional-like value (DR = 5). This is observed when only one BOX is open, either for the monoBOX species (C→O) or for the diBOX (CC→CO) and triBOX (CCC→CCO) compounds, i.e., when the remaining BOXs stay closed. The next BOX openings have much different effects. For the diBOXs, the second opening (CO→OO) is associated with a decrease of β, and this decrease is tuned by controlling the conformation of the π-linker, i.e., the centrosymmetry of the whole compound because β vanishes in centrosymmetric compounds. For the triBOXs, the second opening gives rise to a Λ-shape compound, with a negligible change of β, but a decrease of the DR whereas, along the third opening, β remains similar and the DR decreases to the typical value of octupolar systems (DR = 1.5).     

Growth Rates and Explosions in Sandpiles

We study the abelian sandpile growth model, where n particles are added at the origin on a stable background configuration in ? ^d . Any site with at least 2d particles then topples by sending one particle to each neighbor. We find that with constant background height h≤2d?2, the diameter of the set of sites that topple has order n ^1/d . This was previously known only for h<d. Our proof uses a strong form of the least action principle for sandpiles, and a novel method of background modification. We can extend this diameter bound to certain backgrounds in which an arbitrarily high fraction of sites have height 2d?1. On the other hand, we show that if the background height 2d?2 is augmented by 1 at an arbitrarily small fraction of sites chosen independently at random, then adding finitely many particles creates an explosion (a sandpile that never stabilizes).     

A Combined Kinetic and Thermodynamic Approach for the Interpretation of Continuous‐Flow Heterogeneous Catalytic Processes

The heterogeneous proline‐catalyzed aldol reaction was investigated under continuous‐flow conditions by means of a packed‐bed microreactor. Reaction‐progress kinetic analysis (RPKA) was used in combination with nonlinear chromatography for the interpretation, under synthetically relevant conditions, of important mechanistic aspects of the heterogeneous catalytic process at a molecular level. The information gathered by RPKA and nonlinear chromatography proved to be highly complementary and allowed for the assessment of optimal operating variables. In particular, the determination of the rate‐determining step was pivotal for optimizing the feed composition. On the other hand, the competitive product inhibition was responsible for the unexpected decrease in the reaction yield following an apparently obvious variation in the feed composition. The study was facilitated by a suitable 2D instrumental arrangement for simultaneous flow reaction and online flow‐injection analysis.     

Twisted Poincaré Duality for some Quadratic Poisson Algebras

We exhibit a Poisson module restoring a twisted Poincaré duality between Poisson homology and cohomology for the polynomial algebra endowed with Poisson bracket arising from a uniparametrised quantum affine space. This Poisson module is obtained as the semiclassical limit of the dualising bimodule for Hochschild homology of the corresponding quantum affine space. As a corollary we compute the Poisson cohomology of R, and so retrieve a result obtained by direct methods (so completely different from ours) by Monnier. This research of the first author was supported by a Marie Curie Intra-European Fellowship within the 6th European Community Framework Programme. The second author was supported by EPSRC Grant EP/D034167/1.