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.     

Synthesis and structure‐activity relationship of several aromatic ketone‐based two‐photon initiators

Several novel aromatic ketone‐based two‐photon initiators containing triple bonds and dialkylamino groups were synthesized and the structure‐activity relationships were evaluated. Branched alkyl chains were used at the terminal donor groups to improve the solubility in the multifunctional monomers. Because of the long conjugation length and good coplanarity, the evaluated initiators showed large two‐photon cross section values, while their fluorescence lifetimes and quantum yields strongly depend on the solvent polarity. All novel initiators exhibited high activity in terms of two‐photon‐induced microfabrication. This is especially true for fluorenone‐based derivatives, which displayed much broader processing windows than well‐known highly active initiators from the literature and commercially available initiators. While the new photoinitiators gave high reactivity in two‐photon‐induced photopolymerization at concentration as low as 0.1% wt, these compounds are surprisingly stable under one photon condition and nearly no photo initiation activity was found in classical photo DSC experiment. © 2011 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2011     

Vinylcarbonates and vinylcarbamates: Biocompatible monomers for radical photopolymerization

The last decade has seen a remarkable interest in the use of biocompatible and biodegradable polymers as scaffolds for tissue engineering. The fabrication of 3D scaffolds by lithography‐based additive manufacturing technology (AMT) represents an appealing approach. As poly(lactic acid), the state of the art biocompatible and biodegradable material, cannot be processed by these photopolymerization‐based techniques, it has so far been necessary to use selected (meth)acrylates. By developing new photopolymers based on vinyl carbonates and vinyl carbamates as a reactive group we have been able to avoid most of the disadvantages of classical (meth)acrylate‐based photopolymers. The new generation of biocompatible monomers show low cytotoxicity, have good storage stability, and are sufficiently photoreactive to be structured by lithography based AMT. The mechanical properties and rates of degradation of the polymers can be easily tuned over a broad range. Degradation results in the formation of nonacidic and nontoxic degradation products of low molecular weight that can be easily transported within the human body. Initial in vivo tests showed significant osseointegration of the 3D cellular scaffolds and no signs of implant rejection. © 2010 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2011     

Effect of particle agglomeration and interphase on the glass transition temperature of polymer nanocomposites

In this article, we utilize finite element modeling to investigate the effect of nanoparticle agglomeration on the glass transition temperature of polymer nanocomposites. The case of an attractive interaction between polymer and nanofiller is considered for which an interphase domain of gradient properties is developed. This model utilizes representative volume elements that are created and analyzed with varying degrees of nanoparticle clustering and length scale of interphase domain. The viscoelastic properties of the composites are studied using a statistical approach to account for variations due to the random nature of the microstructure. Results show that a monotonic increase in nanofiller clustering not only results in the loss of interphase volume but also obstructs the formation of a percolating interphase network in the nanocomposite. The combined impacts lead to a remarkable decrease of T_g enhancement of clustering nanofillers in comparison with a well‐dispersed configuration. Our simulation results provide qualitative support for experimental observations that clustering observed at high nanofiller concentrations negatively impacts the effects of the nanofiller on overall properties. © 2011 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2011     

Azido-, hydroxo-, and oxo-bridged copper(II) dimers: spin population analysis within broken-symmetry, density functional methods

Within the general context of homometallic spin-coupled Cu(II) dimers, we propose to relate the antiferromagnetic part of the exchange coupling constant, J _AF, to the quantity ΔP ²(Cu), the difference of copper squared spin populations as calculated for the high-spin (i.e. triplet) and broken-symmetry spin states, through J _AF≈−UΔP ²(Cu), where U is interpreted as the covalent–ionic term. This proportionality is illustrated for three “bare” Cu(II) dimers (i.e. without peripheral ligation, so as to enhance the antiferromagnetic contribution) bridged by azido, hydroxo, or oxo groups. This provides an alternative quantifier of the exchange phenomenon to that usually used, i.e. Δ², the square of the singly occupied molecular orbital splitting in the triplet state. Moreover, and quite interestingly, the quantity ΔP ²(Cu) can become negative (i.e. induce ferromagnetism) without apparently affecting the proportionality relation. Received: 17 September 1999 / Accepted: 9 March 2000 / Published online: 21 June 2000     

Simultaneous reversible addition fragmentation chain transfer and ring‐opening polymerization

The simultaneous ring‐opening polymerization (ROP) of ε‐caprolactone (ε‐CL) and 2‐hydroxyethyl methacrylate (HEMA) polymerization via reversible addition fragmentation chain transfer (RAFT) chemistry and the possible access to graft copolymers with degradable and nondegradable segments is investigated. HEMA and ε‐CL are reacted in the presence of cyanoisopropyl dithiobenzoate (CPDB) and tin(II) 2‐ethylhexanoate (Sn(Oct)₂) under typical ROP conditions (T > 100 °C) using toluene as the solvent in order to lead to the graft copolymer PHEMA‐g‐PCL. Graft copolymer formation is evidenced by a combination of size‐exclusion chromatography (SEC) and NMR analyses as well as confirmed by the hydrolysis of the PCL segments of the copolymer. With targeted copolymers containing at least 10% weight of PHEMA and relatively small PHEMA backbones (ca. 5,000–10,000 g mol^?1) the copolymer grafting density is higher than 90%. The ratio of free HEMA‐PCL homopolymer produced during the “one‐step” process was found to depend on the HEMA concentration, as well as the half‐life time of the radical initiator used. © 2008 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 46: 3058–3067, 2008     

Global G-Manifold Reductions and Resolutions

The purpose of this note is to exhibit some simple and basic constructions for smooth compact transformation groups, and some of their most immediate applications to geometry.