The asymptotic expansion for n! and the Lagrange inversion formula

We obtain an explicit simple formula for the coefficients of the asymptotic expansion for the factorial of a natural number, $$n!=n^n\sqrt{2\pi n}\mbox{e}^{-n}\biggl\{1+\frac{a_1}{n}+\frac{a_2}{n^2}+\frac{a_3}{n^3}+\cdots\biggr\},$$ in terms of derivatives of powers of an elementary function that we call normalized left truncated exponential function. The unique explicit expression for the a _k that appears to be known is that of Comtet in (Advanced Combinatorics, Reidel, 1974), which is given in terms of sums of associated Stirling numbers of the first kind. By considering the bivariate generating function of the associated Stirling numbers of the second kind, another expression for the coefficients in terms of them follows also from our analysis. Comparison with Comtet??s expression yields an identity which is somehow unexpected if considering the combinatorial meaning of the terms. It suggests by analogy another possible formula for the coefficients, in terms of a normalized left truncated logarithm, that in fact proves to be true. The resulting coefficients, as well as the first ones are identified via the Lagrange inversion formula as the odd coefficients of the inverse of a pair of formal series. This in particular leads to the identification of a couple of simple implicit equations, which permits us to obtain also some recurrences related to the a _k ??s.     

Iron substitution effects and Mössbauer spectroscopy in Bi−Sr−Ca−Cu ceramic oxides

Mössbauer spectroscopy spectra of the ceramic Bi₂Sr₂Ca(Cu_1-x Fe _x )₂O_8+y forx=0, 0.005, 0.02 and 0.05 shows two doublets, that can be assigned to two different environments for the substituting ion. One of the sites is identified as a single Fe ion replacing a Cu ion in the CuO plane. The other site is associated with the dopant in an interstitial position in the plane, in slightly different environments. This last configuration, which increases with increasingx, is related with adverse effects upon the conductive properties of these materials.     

EPMA analysis of float glass surfaces

This work concentrates on the setting-up of conditions for the quantitative analysis of the composition profiles of major and minor components of float glass surfaces with an electron probe micro-analyser. Since surfaces are modified for depths of 10–50 m, they have been studied both in polished cross-section and perpendicularly to the electron beam by varying its energy to investigate the near-surface region (0–3m). The results demonstrate that combined use of the two methods supplies complete and continuous information on the quantitative distribution from large depths to the first layers of the float surfaces. The experimental conditions, sample preparation, in-depth and lateral resolutions, precision and accuracy of the methods are also investigated.     

Mean field like renormalization group for disordered systems

Mean field results for small clusters of spins are combined with renormalization group ideas to give a new approximate scheme for the study of disordered systems. Dilute, random fields and random bonds Ising systems on a d-dimensional hypercubic lattice are analyzed with this new scheme.     

Synthesis and morphology of model 3‐miktoarm star terpolymers of styrene,isoprene and 2‐vinyl pyridine

The combination of anionic polymerization and controlled chlorosilane chemistry made possible for the first time the synthesis of model 3‐miktoarm star terpolymers of styrene (PS), isoprene (PI) and 2‐vinylpyridine (P2VP) (3μ‐SIV). The morphology of a nearly symmetric 3μ‐SIV star terpolymer, was also studied. From the preliminary results, it seems that the PI and P2VP phases form hexagonally packed adjoined cylinders, whereas the PS phase occupies the remaining space forming non‐regular curved hexagons, hexagonally packed as well. The star junction points reside on periodically spaced, parallel lines defined by the intersection of the three microdomain interfaces. Non of the phases form the matrix. The star molecular architecture gives the molecule the ability to “choose” which arms directly interact in the microphase segregate state, in order to minimize the most highly unfavorable contact between the PI and P2VP arms.     

Characterization of polypropylene (PP) nanocomposites for industrial applications

This study aims in the examination of a new class of materials named polymer layered silicate nanocomposites. In our case, composites are usually combinations of polypropylene matrix with solid mineral reinforcements named silicates (e,g. montmorillonite, a natural clay). In this study, two complementary techniques used to characterize nanocomposites. Fourier transform infrared spectroscopy (FT-IR) both in transmission and attenuated total reflectance (ATR) modes combined with X-ray photoelectron spectroscopy (XPS).     

Characterization of hydrophilic networks synthesized by group transfer polymerization

Group transfer polymerization was used to synthesize several series of hydrophilic random and model networks. Cationic random networks were prepared both in bulk and in tetrahydrofuran (THF) using a monofunctional initiator and simultaneous polymerization of monomer and branch units, while a bifanctional initiator was employed in THF for the synthesis of model networks comprising basic or acidic chains. Upon polymerization of the monomer, the latter initiator gives linear polymer chains with two “living” ends, which are subsequently interconnected to a polymer network by the addition of a branch unit. Homopolymer network star polymers were also synthesized in THF by a one‐pot procedure. The synthesis involved the use of a monofunctional initiator and the four‐step addition of the following reagents: (i) monomer, to give linear homopolymers; (ii) branch unit, to form “arm‐first” star polymers; (iii) monomer, to form secondary arms and give “in‐out” star polymers; and, finally (iv) branch unit again, to interconnect the “in‐out” stars to networks. Different networks were prepared for which the degree of polymerization (DP) of the linear chains between junction points was varied systematically. For all networks synthesized, the linear segments, the “arm‐first” and the “in‐out” stars were characterized in terms of their molecular weight (MW) and molecular weight distribution (MWD) using gel permeation chromatography (GPC). The degrees of swelling of both the random and model networks in water were measured and the effects of DP, pH, and monomer type were investigated.