Rigorous results on the thermodynamics of the dilute Hopfield model

We study the Hopfield model of an autoassociative memory on a random graph onN vertices where the probability of two vertices being joined by a link isp(N). Assuming thatp(N) goes to zero more slowly thanO(1/N), we prove the following results: (1) If the number of stored patternsm(N) is small enough such thatm(N)/Np(N) 0, asN, then the free energy of this model converges, upon proper rescaling, to that of the standard Curie-Weiss model, for almost all choices of the random graph and the random patterns. (2) If in additionm(N) < ln N/ln 2, we prove that there exists, forT< 1, a Gibbs measure associated to each original pattern, whereas for higher temperatures the Gibbs measure is unique. The basic technical result in the proofs is a uniform bound on the difference between the Hamiltonian on a random graph and its mean value.     

Conformational statistics of polymer chain terminally attached to wall (Ⅲ)——NRW model loop chain

When the two end groups of a linear polymer chain are absorbed on a solid surface,the polymer chain forms the "loop" conformation.Investigation has been made on the conformational statistics of a model loop chain by the normal landom walk (NRW) on a lattice confined in the half-infinite space.Based on the conformational distribution function of the NRW model tail chain,it is easy to deduce an analytical formula expressing the conforma-tional number of the model loop chain.It was found that the ratio of the conformational number of the model loop chain to that of the free chain varies with the power function N-2/3 when the chain length N→∞ The same result was obtained by means of the recursion equation.The ratio of the mean square end-to-end distance h2 for the model loop chain to its mean square bond length I2 is 2N/3 Compared with the free chain with the same length N,the mean square end-to-end distance of the model loop chain contracts to a certain extent.The basic relationships deduced were support     

Fractal behavior in trapping and reaction: A random walk study

We investigate the trapping of a random walker in fractal structures (Sierpinski gaskets) with randomly distributed traps. The survival probability is determined from the number of distinct sites visited in the trap-free fractals. We show that the short-time behavior and the long-time tails of the survival probability are governed by the spectral dimensiond. We interpolate between these two limits by introducing a scaling law. An extension of the theory, which includes a continuous-time random walk on fractals, is discussed as well as the case of direct trapping. The latter case is shown to be governed by the fractal dimensiond.     

The mean spherical model in a random external field and the replica method

We provide a quick elementary solution of the mean spherical model in a random external field. This also allows an immediate proof of the self-averaging property of the free energy. We calculate the free energy by means of the replica method, i.e., for any (not necessarily integer) replica numbern, and show that when a phase transition occurs the limits andn 0 are not interchangeable.     

Quasi‐random integration in quantum chemistry: Efficiency,stability, and application to the study of small atoms and molecules constrained in spherical boxes

This project consists of two parts. In the first part, a series of test calculations is performed to verify that the integrals involved in the determination of atomic and molecular properties by standard self‐consistent field (SCF) methods can be obtained through Halton, Korobov, or Hammersley quasi‐random integration procedures. Through these calculations, we confirm that all three methods lead to results that meet the levels of precision required for their use in the calculation of properties of small atoms or molecules at least at a Hartree–Fock level. Moreover, we have ensured that the efficiency of quasi‐random integration methods that we have tested is Halton=Korobov>Hammersley?pseudo‐random. We also find that these results are comparable to those yielded by ordinary Monte Carlo (pseudo‐random) integration, with a calculation effort of two orders of smaller magnitude. The second part, which would not have been possible without the integration method previously analyzed, contains a first study of atoms constrained in spherical boxes through SCF calculations with basis functions adapted to the features of the problem: Slater‐type orbitals (STOs) trimmed by multiplying them by a function that yields 1 for 0 < r < (R‐δ), polynomial values for (R‐δ) < r < R and null for r > R, R being the radius of the box and δ a variationally determined interval. As a result, we obtain a equation of state for electrons of small systems, valid just in the limit of low temperatures, but fairly simple. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Derivation of the transport equation for electrons moving through random impurities

A single (nonrelativistic, spinless) electron subject to a constant external electric field interacts with impurities located on an infinitely extended lattice by a potential of random strength. The random strength is given by a field of Gaussian random variables. We show the existence of the averaged dynamics and prove that in the weak coupling limit, 0, ₂ t= fixed, one obtains the usual transport equation for the velocity distribution.Work supported by a Max Kade Foundation fellowship.On leave of absence of the Fachbereich Physik der Universität München.     

Effects of processing parameters on the mechanical properties of polypropylene random copolymer

The mechanical properties of polypropylene random copolymer (PP-R) with different processing parameters were studied. Special attention is devoted to the investigation of the influence of masterbatch addition on the variation in the mechanical properties of injection moulded PP-R. Tensile, instrumented Charpy impact, Shore D hardness, differential scanning calorimeter (DSC) and Vicat softening temperature (VST) tests were conducted on the test samples containing different colour masterbatches varying from 0.5 to 10 wt%. The observed changes in the mechanical behaviour are explained by the type and level of masterbatch content. The natural UV weathering performance of the PP-R material was studied from the masterbatch type point of view. The effect of processing parameters on material performance was studied on samples which were directly obtained from extruded pipes and on injection moulded samples.

Finally, the effects of storage time on the polymer properties were investigated.     

Cationic copolymerization of indene with styrene derivatives: Synthesis of random copolymers of indene with high molecular weight

Random copolymers with high molecular weights of indene and p‐methylstyrene (pMeSt) were synthesized by cationic polymerization with trichloroacetic acid/tin tetrachloride in CH₂Cl₂ at low temperatures. When indene and pMeSt (1:1 v/v), for example, were polymerized at ?40 °C, both monomers were consumed at very similar rates to give a copolymer with high molecular weight [number‐average molecular weight (M_n): 8–9 × 10⁴]. This is indeed quite unexpected behavior for the combination of these two monomers because pMeSt polymerized over 1000 times faster than indene in the homopolymerization under the reaction conditions previously described. The product copolymer of indene and pMeSt had a random monomer sequence in it that was confirmed by NMR analyses and thermal‐property measurements. In sharp contrast with pMeSt, styrene and p‐chlorostyrene, which have no electron‐donating groups on the phenyl ring, led to low molecular weight polymers (M_n < 10,000) in the copolymerization with indene (1:1 v/v). © 2002 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 40: 2449–2457, 2002     

Microstructure and ferroelectric properties of r.f. magnetron sputtering derived PZT thin films deposited on interlayer (PbO/TiO2)

Lead zirconate titanate (PZT) thin films were deposited on Pt/Ti/SiO₂/Si and interlayer/Pt/Ti/SiO₂/Si substrate by radio frequency (r.f.) magnetron sputtering with a Pb_1.1Zr_0.53Ti_0.47O₃ target. The crystallization of the PZT thin films was formed only by substrate temperature. When interlayer (PbO/TiO₂) was inserted between the PZT thin film and the Pt electrode, the grain growth and processing temperature of the PZT thin films were considerably improved. Compared to PZT/Pt structure, the dielectric constant and polarization properties of the PZT/interlayer/Pt structure were fairly improved. In particular, PZT/interlayer/Pt at the substrate temperature of 400 °C showed prevalent ferroelectric properties (_r=475.97, tanδ=0.0591, P_r=23 μC/cm²). As a result of an X-ray photoelectron spectroscopy (XPS) depth-profile analysis, it was found that PZT/interlayer/Pt deposited only by substrate temperature without the post-annealing process via r.f. magnetron sputtering method remained independent of each other regardless of substrate temperatures.     

The size distribution for theA g RB f−g Model of polymerization

This paper gives the equilibrium distribution of polymer sizes for Flory'sA _g RB _f–g model of polymerization. In this model, the polymers are composed of structural units withg functional groups of the typeA and (f-g) functional groups of the typeB. Reaction is subject to three conditions: (1) Functional groups of the typeA react only with those of typeB, and vice versa. (2) Intramolecular reactions do not occur [and therefore only branched-chain (noncyclic) polymers and formed]. (3) Subject to conditions (1) and (2), all functional groups are equally reactive. The derivation employs Stockmayer's statistical mechanical method (first used on Flory'sRA _f model), coupled with a recursion giving the number of distinct polymers which may be assembled fromk units of theA _g RB _f–g type. We also give distributions for a limiting case of theA _g RB _f–g model, the so-calledA _g RB model. This paper completes the solution of the Smoluchowski coagulation equation (monodisperse case) for the kernelsa _ij =A + B(i +j)+ Cij. The proof will be given in another publication.