Some functional equations originating from number theory

We will introduce new functional equations (3) and (4) which are strongly related to well-known formulae (1) and (2) of number theory, and investigate the solutions of the equations. Moreover, we will also study some stability problems of those equations.     

Quantitative assessment of solid oxide electrochemical doping

Quantitative analysis of metal cation doping by solid oxide electrochemical doping (SOED) has been performed under galvanostatic doping conditions. A M–β″-Al₂O₃ (M=Ag, Na) microelectrode (contact radius: about 10 μm) was used as cation source to attain a homogeneous solid–solid contact between the β″-Al₂O₃ and doping target. In Ag doping into alkali borate glass, the measured dopant amount closely matched the theoretical value. High Faraday efficiencies of above 90% were obtained. This suggests that the dopant amount can be precisely controlled on a micromole scale by the electric charge during electrolysis. On the other hand, current efficiencies of Na doping into Bi₂Sr₂CaCu₂O_y (BSCCO) ceramics depended on the applied constant current. Efficiencies of above 80% were achieved at a constant current of 10 μA (1.6 A cm⁻²). The relatively low efficiencies were explained by the saturation of BSCCO grain boundaries with Na. By contrast, excess Na was detected on the anodic surface of ceramics at a constant current of 100 μA (16 A cm⁻²). In the present study, we demonstrate that SOED enables micromole-scale control over dopant amount.     

Flat rank of automorphism groups of buildings

The flat rank of a totally disconnected locally compact group G, denoted flat-rk(G), is an invariant of the topological group structure of G. It is defined thanks to a natural distance on the space of compact open subgroups of G. For a topological Kac-Moody group G with Weyl group W, we derive the inequalities alg-rk(W) ≤ flat-rk(G) ≤ rk(|W|₀). Here, alg-rk(W) is the maximal Z-rank of abelian subgroups of W, and rk(|W|₀) is the maximal dimension of isometrically embedded flats in the CAT0-realization |W|₀. We can prove these inequalities under weaker assumptions. We also show that for any integer n ≥ 1 there is a simple, compactly generated, locally compact, totally disconnected group G, with flat-rk(G) = n and which is not linear.     

Automated derivatization for on-line solid-phase extraction-gas chromatography; phenolic compounds

An automated system for derivaatization was coupled on-line with solid-phase extraction-gas-chromatography (SPE-GC). The system was optimized for the determination of phenol and chlorinated phenols in aqueous samples. The test analytes were acetylated with acetic anhydride; proper buffering of the sample was a critical factor. Next, the phenol acetates were enriched on a SPE cartridge and transferred to a GC; two appraoaches were studied. In the first approach, the derivatives were enriched on disposable C18 cartridges (ASPEC type) and desorbed with methylacetate. Aan aliquot of the final eluate was injected on-line the GC by means of a loop-type interface. In the second approach, trace enrichment was performed on 10 × 2 mm i.d. LC-type precolumn packed with polystyrenedivinylbenzene copolymer (PLRP-S) this precolumn was dried with a mitrogen purge and the phenol acetates were desorbed with ethyl acetate which was injectedon-line into the retention gap of the GC under partially concurrent solvent evaporation (PCSE) conditions. The Derivatization-SPE-GC system which was based on the loop-type interface has the advantage of simplicity and easy operation, the main drawback is the impossibility to determine phenol acetates which elute prior to trichlorophenol acetates. With the derivatization-SPE-GC approach using PCSE-based desorption, even the most volatile analyte of the test series, phenol acetate, can be determined successfully. The entire procedure, including the derivatization step, was fully automated and integrated in one set-up. The precision data for the integrated on-line derivatization-SP-FID system were fully satisfactory, with RSD values of 1–12 % at the 1 μg/1 level. When a sample volume of 2.2 ml was analyzed, The detection limits for the chlorinated phenol acetates were in the 0.1–0.3 μg/1 range.     

Evaluation of compatibility in polymer blend systems by simultaneous DSC-FTIR measurement

The concept of crystallization dynamics method evaluating the miscibility of binary blend system including crystalline component was proposed. Three characteristic rates, nucleation, crystal growth rates (N*, G*) and growth rate of conformation (G _c*) were used to evaluate the miscibility of PVDF/at-PMMA and PVDF/iso-PMMA by the simultaneous DSC-FTIR. N*, G* and G _c* depended remarkably on both temperature and blend fraction (ϕ_PMMA) for PVDF/at-PMMA system, which indicated the miscible system. PVDF/iso-PMMA showed small ϕ_PMMA dependency of N*, G* and G _c*, was estimated the immiscible system. The ΔT/T _m⁰ values, corresponding to Gibbs energy required to attend the constant G* and G _c*, evaluated from G* and G _c* showed the good linear relationships with different slope. The experimental results suggested that the concentration fluctuation existed in PVDF/iso-PMMA system.     

Characteristics of Ti0.97Co0.03O2:Sb0.01 dilute magnetic semiconducting thin films deposited at 500 °C by pulsed laser deposition

Epitaxial Ti_0.97Co_0.03O₂:Sb_0.01(TCO:Sb) films were deposited on R-Al₂O₃ (1 1 0 2) substrates at 500 °C in various deposition pressures by pulsed laser deposition. The solubility of cobalt within the films increases with decreasing deposition pressure at a deposition temperature of 500 °C. The TCO:Sb films deposited at 5×10⁻⁶ Torr exhibit a p-type anomalous Hall effect having a hole concentration of 6.1×10²²/cm³ at 300 K. On the other hand, films deposited at 4×10⁻⁴ Torr exhibits an n-type anomalous Hall effect having an electron concentration of about 1.1×10²¹/cm³. p- or n-type DMS characteristics depends on the change of the structure of TCO:Sb films and the solubility of Co is possible by controlling the deposition pressure.     

Badge-type diffusive sampler using 3-methyl-2-benzothiazolinone hydrazone for measuring formaldehyde in indoor air.

The evaluation of a badge-type diffusive sampler for measuring formaldehyde using 3-methyl-2-benzothiazolinone hydrazone (MBTH) was investigated. On average, the formaldehyde concentration in blanks was reduced by approximately 31% by cleaning procedures. The cleaning techniques did not significantly differ in effectiveness. The maximum sampling rate was 22.4 +/- 3.5 mL min(-1) at MBTH concentrations of 0.05%. The formaldehyde concentration in blanks did not appreciably increase over a period of about 1 month at room temperature, and was 0.36 +/- 0.03 microg, with a relative standard deviation of 8%. The diffusive sampler had good precision and accuracy for measuring formaldehyde in indoor environments. For a 24-h exposure time, the limits of detection and quantification calculated with the field blanks were 9.7 and 13.8 ppb, respectively. The minimum exposure times were calculated based on the measured and calculated limits of quantification, the sampling rate, and the atmospheric formaldehyde concentration. The capacity of the diffusive sampler with 0.5% MBTH was 3 ppm h(-1), approximately 1.5-times the capacity when the MBTH concentrations were 0.05%.