A Note on Log-Convexity of q-Catalan Numbers

The q-Catalan numbers studied by Carlitz and Riordan are polynomials in q with nonnegative coefficients. They evaluate, at q = 1, to the Catalan numbers: 1, 1, 2, 5, 14,…, a log-convex sequence. We use a combinatorial interpretation of these polynomials to prove a q-log-convexity result. The sequence of q-Catalan numbers is not q-log-convex in the narrow sense used by other authors, so our work suggests a more flexible definition of q-log convex be adopted. Received January 2, 2007     

Complexes in polymers IV. FTIR microscopic characterization of thin organic polymer films containing embedded transition-metal carbonyl complexes

Thin films of polystyrene (PS), poly(methyl methacrylate) (PMMA) and polystyrene-polyacrylonitrile copolymer (PS-AN), containing various embedded transition-metal complexes, have been studied by FTIR microscopy. The spatial distributions of the transition-metal carbonyl complexes throughout the thin organic polymer films have been determined by a two-dimensional IR mapping procedure. The spectral variations observed in the distribution of the metal carbonyls throughout the different polymer films are discussed. The IR data show that the technique used to prepare the organometallic-embedded thin films (viz. freeze-drying of solutions followed by hot mechanical pressing of the residues) does in general lead to homogeneous films which may eventually find industrial application, e.g. as membrane sensors for small molecules.     

Metal–Nitrosyl complexes as a source of new vasodilators: Strategies derived from systematic chemistry and nitrosyl ligand reactivity

A series of nitrosyl complexes of empirical formula K_n[M(CN)₅NO], where M = V, Cr, Mn and Co and n = 3, or M = Mo and n = 4, have been prepared which are notional analogues of the widely used vasodilator sodium nitroprusside. Their reactivity towards common nucleophiles (OH^?, NH₂R, NHR₂, HS^? and RS^?), acid and photolysis has been investigated to elucidate the desired properties required of new metal nitrosyls which may have some potential as new non-cyanide-based vasodilators.     

Modification of magnetic anisotropy in metallic glasses using high-energy ion beam irradiation

Heavy ion irradiation in the electronic stopping power region induces macroscopic dimensional change in metallic glasses and introduces magnetic anisotropy in some magnetic materials. The present work is on the irradiation study of ferromagnetic metallic glasses, where both dimensional change and modification of magnetic anisotropy are expected. Magnetic anisotropy was measured using Mössbauer spectroscopy of virgin and irradiated Fe₄₀Ni₄₀B₂₀ and Fe₄₀Ni₃₈Mo₄B₁₈ metallic glass ribbons. 90 MeV ¹²⁷I beam was used for the irradiations. Irradiation doses were 5×10¹³ and 7.5×10¹³ ions/cm². The relative intensity ratios D ₂₃ of the second and third lines of the Mössbauer spectra were measured to determine the magnetic anisotropy. The virgin samples of both the materials display in-plane magnetic anisotropy, i.e., the spins are oriented parallel to the ribbon plane. Irradiation is found to cause reduction in magnetic anisotropy. Near-complete randomization of magnetic moments is observed at high irradiation doses. Correlation is found between the residual stresses introduced by ion irradiation and the change in magnetic anisotropy.     

Rook theory and cycle-counting permutation statistics

A statistic is found to combinatorially generate the cycle-counting q-hit numbers, defined algebraically by Haglund [Adv. in Appl. Math. 17 (1996) 408–459]. We then define the notion of a cycle-Mahonian pair of statistics (generalizing that of a Mahonian statistic), and show that our newly discovered statistic is part of such a pair. Finally, we note a second example of a cycle-Mahonian pair of statistics which leads us to define the stronger property of being a cycle-Euler–Mahonian pair.     

Observation of Substitutional Fe in CEMS Measurements on Synthetic CVD Diamond

Hyperfine Interactions - Conversion electron Mössbauer spectroscopy measurements have been made on a diamond sample synthesized by chemical vapour deposition. The sample was implanted with 70...     

Orientation and relaxation of polymer–clay solutions studied by rheology and small-angle neutron scattering

The influence of shear on viscoelastic solutions of poly(ethylene oxide) (PEO) and clay [montmorillonite, i.e., Cloisite NA+ (CNA)] was investigated with rheology and small-angle neutron scattering (SANS). The steady-state viscosity and SANS were used to measure the shear-induced orientation and relaxation of the polymer and clay platelets. Anisotropic scattering patterns developed at much lower shear rates than in pure clay solutions. The scattering anisotropy saturated at low shear rates, and the CNA clay platelets aligned with the flow, with the surface normal parallel to the gradient direction. The cessation of shear led to partial and slow randomization of the CNA platelets, whereas extremely fast relaxation was observed for laponite (LRD) platelets. These PEO–CNA networklike solutions were compared with previously reported PEO–LRD networks, and the differences and similarities, with respect to the shear orientation, relaxation, and polymer–clay interactions, were examined. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 3102–3112, 2004