Expected Sums of General Parking Functions

A (u₁, u₂, . . . )-parking function of length n is a sequence (x₁, x₂, . . . , x_n) whose order statistics (the sequence (x₍₁₎, x₍₂₎, . . . , x_(n)) obtained by rearranging the original sequence in non-decreasing order) satisfy x_(i) u_(i). The Gonarov polynomials g _n (x; a₀, a ₁, . . . , a _n-1) are polynomials biorthogonal to the linear functionals (a _i) Dⁱ, where (a) is evaluation at a and D is differentiation. In this paper, we give explicit formulas for the first and second moments of sums of u-parking functions using Gonarov polynomials by solving a linear recursion based on a decomposition of the set of sequences of positive integers. We also give a combinatorial proof of one of the formulas for the expected sum. We specialize these formulas to the classical case when u _i=a+ (i-1)b and obtain, by transformations with Abel identities, different but equivalent formulas for expected sums. These formulas are used to verify the classical case of the conjecture that the expected sums are increasing functions of the gaps u_i+1 - u_i. Finally, we give analogues of our results for real-valued parking functions.AMS Subject Classification: 05A15, 05A19, 05A20, 05E35.     

Detecting positive feedback in multivariate time series: The case of metal prices and US inflation

The objective of this paper is to examine causality and feedback relationships between primary commodity prices and US inflation. To this end, the bivariate noisy Mackey–Glass process recently developed by Kyrtsou and Labys [Evidence for chaotic dependence between US inflation and commodity prices, J. Macroecon. 28(1) (2006) 256–266] has been applied to assess this relationship. Results obtained support evidence in favour of causality, which can help to identify the influences of speculative price behaviour on inflation.     

Thermal stability of alkoxy monolayers grafted on Si(1 1 1)

Direct grafting of organic monolayers on Si is of prime interest in order to give specific properties to a silicon surface. However, for microelectronics applications, this possibility is hampered by the limited stability of the grafted layers. It has been previously established that alkyl layers attached to Si surfaces through Si-C bonds become unstable at 250-300 °C, by desorption of alkenes. Changing the nature of the bonding to the surface might allow one to circumvent this desorption pathway and increase the layer stability. In our work, decanol and decyl aldehyde are reacted with the Si(1 1 1)-H surface at ∼100 °C during 20 h in order to obtain alkoxy monolayers. FTIR measurements performed in ATR geometry show that the grafted molecule surface coverage is on the order of 33% after reaction with decanol and 50% after reaction with decyl aldehyde. Characterization by AFM essentially reveals that the morphology of the grafted surfaces is unaffected as compared to that of Si-H surfaces. However, the edges of the terraces at alcohol-grafted surfaces exhibit some pitting, probably due to the presence of water in the grafting liquid. Thermal stability studies show that alkoxy chains progressively disappear from the Si surface between 200 and 400 °C. From the CH₂/CH₃ ratio in the CH region (2760-3070 cm⁻¹), it appears that the chains undergo progressive dissociation by C-C bond breaking before their complete disappearance from the surface. Therefore, the thermal behaviour of alkoxy monolayers appears quite distinct from that of alkyl monolayers that tend to leave the surface in a much narrower temperature range (250-350 °C), essentially via breaking of the Si-C bonds.     

N-3 alkylation of uracils with unprotected amino alcohols using the Mitsunobu reaction

During our search for novel CGRP antagonists, we had great difficulty in accessing one of our key motifs. Herein, we communicate how we solved the problem by an unprecedented Mitsunobu alkylation using unprotected amino alcohols.     

Energy transfer (EnT) photocatalysis enabled by gold-N-heterocyclic carbene (NHC) complexes

We present the use of gold sensitizers [Au(SIPr)(Cbz)] (PhotAu 1) and [Au(IPr)(Cbz)] (PhotAu 2) as attractive alternatives to state-of-the-art iridium-based systems. These novel photocatalysts are deployed in [2 + 2] cycloadditions of diallyl ethers and N-tosylamides. The reactions proceed in short reaction times and in environmentally friendly solvents. [Au(SIPr)Cbz] and [Au(IPr)(Cbz)] have higher triplet energy (E_T) values (66.6 and 66.3 kcal mol⁻¹, respectively) compared to commonly used iridium photosensitizers. These E_T values permit the use of these gold complexes as sensitizers enabling energy transfer catalysis involving unprotected indole derivatives, a substrate class previously inaccessible with state-of-the-art Ir photocatalysts. The photosynthesis of unprotected tetracyclic spiroindolines via intramolecular [2 + 2] cycloaddition using our simple mononuclear gold sensitizer is readily achieved. Mechanistic studies support the involvement of triplet–triplet energy transfer (TTEnT) for both [2 + 2] photocycloadditions.

We present the use of gold sensitizers [Au(SIPr)(Cbz)] (PhotAu 1) and [Au(IPr)(Cbz)] (PhotAu 2) as attractive alternatives to state-of-the-art iridium-based systems.     

In situ single‐crystal to single‐crystal (SCSC) transformation of the one‐dimensional polymer catena‐poly[[diaqua(sulfato)copper(II)]‐μ2‐glycine] into the two‐dimensional polymer poly[μ2‐glycine‐μ4‐sulfato‐copper(II)]

The one‐dimensional coordination polymer catena‐poly[diaqua(sulfato‐κO)copper(II)]‐μ₂‐glycine‐κ²O:O′], [Cu(SO₄)(C₂H₅NO₂)(H₂O)₂]_n, (I), was synthesized by slow evaporation under vacuum of a saturated aqueous equimolar mixture of copper(II) sulfate and glycine. On heating the same blue crystal of this complex to 435 K in an oven, its aspect changed to a very pale blue and crystal structure analysis indicated that it had transformed into the two‐dimensional coordination polymer poly[(μ₂‐glycine‐κ²O:O′)(μ₄‐sulfato‐κ⁴O:O′:O′′:O′′)copper(II)], [Cu(SO₄)(C₂H₅NO₂)]_n, (II). In (I), the Cu^II cation has a pentacoordinate square‐pyramidal coordination environment. It is coordinated by two water molecules and two O atoms of bridging glycine carboxylate groups in the basal plane, and by a sulfate O atom in the apical position. In complex (II), the Cu^II cation has an octahedral coordination environment. It is coordinated by four sulfate O atoms, one of which bridges two Cu^II cations, and two O atoms of bridging glycine carboxylate groups. In the crystal structure of (I), the one‐dimensional polymers, extending along [001], are linked via N—H...O, O—H...O and bifurcated N—H...O,O hydrogen bonds, forming a three‐dimensional framework. In the crystal structure of (II), the two‐dimensional networks are linked via bifurcated N—H...O,O hydrogen bonds involving the sulfate O atoms, forming a three‐dimensional framework. In the crystal structures of both compounds, there are C—H...O hydrogen bonds present, which reinforce the three‐dimensional frameworks.     

Uniqueness and continuous dependence on the initial data for a class of non-linear shallow shell problems

This note is concerned with the non-linear shallow shell model introduced in 1966 by W.T. Koiter, and later studied by M. Bernadou and J.T. Oden. We show the uniqueness of the solution to the dynamical model and that this solution is continuous with respect to the initial data. To cite this article: J. Cagnol et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Study by XPS of the chlorination of proteins aggregated onto tin dioxide during electrochemical production of hypochlorous acid

In solution, hypochlorous acid (HOCl) reacts with organic matter and notably with protein side-chains. In this study, HOCl was produced by an electrochemical way, by oxidation of chloride ions at a transparent tin dioxide electrode in the presence of a protein, the bovine serum albumin (BSA). A thick irregular layer is formed at the electrode when HOCl is produced at the SnO₂ surface. Indeed, SEM analyses show that an important deposit is formed during the anodic polarization of SnO₂ in the presence of chloride ions and proteins. Actually, two phenomena take place on the one hand the chlorination of the proteins due to the reaction of HOCl with some protein side-chains and on the other hand the aggregation of proteins onto the SnO₂ surface. The present X-ray photoelectron spectroscopy study points out the cross-linking of BSA molecules via formation of inter molecular sulfonamide groups. It also shows that the BSA chlorination is due on the one hand to the formation of sulfonyl chloride groups (-SO₂Cl) and on the other hand to formation of chloramine groups (N-Cl). The Cl2p and S2p photo-peak intensities allowed us to quantify the chloramines. It is found that, one BSA entity immobilized onto the SnO₂ surface contains about 50 chloramine groups.     

Isomorphous replacement of M(II) ions in M(II)-Gd(III) dimers (M(II) = Cu(II), Mn(II), Ni(II), Co(II), Zn(II)): magnetic studies of the products

Complexes [M(II)Gd(III){pyCO(OEt)pyC(OH)(OEt)py}?](ClO?)?·EtOH [M(II) = Cu(II) (1), Mn(II) (2), Ni(II) (3), Co(II) (4) and Zn(II) (5)] crystallize in the monoclinic Cc space group and contain one hexacoordinate M(II) ion and one enneacoordinate Gd(III) ion, bridged by three {pyCO(OEt)pyC(OH)(OEt)py}? ligands. Magnetic susceptibility measurements indicate a ferromagnetic interaction for 1 and antiferromagnetic interactions for 2-4. Using the ? = -J?(Gd(III))?(M(II)) spin Hamiltonian formalism, fits to the magnetic susceptibility data yielded J values of +0.32 cm?1 for 1, -1.7 cm?1 for 2, and -0.22 cm?1 for 3. In complex 4, the orbital contributions of Co(II) precluded the determination of the magnetic coupling. The complex follows the Curie-Weiss law with θ = -2.07 K (-1.44 cm?1).