Algebraic methods for the solution of linear functional equations

The equation

$$\sum^{n}_ {i=0} a_{i}f(b_{i}x + (1 - b_{i})y) = 0$$

belongs to the class of linear functional equations. The solutions form a linear space with respect to the usual pointwise operations. According to the classical results of the theory they must be generalized polynomials. New investigations have been started a few years ago. They clarified that the existence of non-trivial solutions depends on the algebraic properties of some related families of parameters. The problem is to find the necessary and sufficient conditions for the existence of non-trivial solutions in terms of these kinds of properties. One of the earliest results is due to Z. Daróczy [1]. It can be considered as the solution of the problem in case of n = 2. We are going to take more steps forward by solving the problem in case of n = 3.

     

Quantitative separation of beryllium from magnesium,calcium, aluminium,iron and other elements by cation-exchange chromatography in nitric acid-methanol

Beryllium is separated from Mg, Ca, Mn(II), Fe(III), Al, Co(II). Zn. U(VI), La and Gd by elution with 2.0 M nitric acid in 70 % methanol from a column of AG50W-X8 sulphonated polystyrene cation exchanger, while the other elements are retained quantitatively. Sr, Ba, Sc, Y, the other lanthanides, Zr, Hf, Th, Ga, In, Cd and Ni(II) should also be separated according to their distribution coefficients or elution behaviour. Separations are sharp and recoveries quantitative from millimolar amounts down to 10 μg of beryllium. The separation of Ti(IV) and Cu(II) from beryllium is not satisfactory and requires rather large columns. Bi(III), Pb(II), Hg(II) and the alkali metals are eluted together with beryllium, but can be separated by other methods. Typical elution curves and results for the quantitative separation of binary synthetic mixtures are presented.     

Preparation and substitution chemistry of [Bu4N]2[W6Cl8(p-OSO2C6H4CH3)6]. A useful precursor for pseudohalide, acetate, and organometallic complexes containing the [W6Cl8](4+) core

The tosylate (p-toluenesulfonate) cluster [Bu4N]2[W6Cl8(p-OSO2C6H4CH3)6] (1) has been prepared and characterized by IR and NMR spectroscopy, elemental analysis, and an X-ray crystal structure. This cluster complex is shown to be a useful starting material for the preparation of pseudohalide clusters, [Bu4N]2[W6Cl8(NCQ)6] (Q = O (2), S (3), and Se (4)), in high yields. Cluster 1 also serves as a precursor to the new cluster compounds: [Bu4N]2[W6Cl8(O2CCH3)6] (5), [Bu4N]2[W6Cl8((mu-NC)Mn(CO)2(C5H5))6] (6), [W6Cl8((mu-NC)Ru(PPh3)2(C5H5))6][ p-OSO2C6H4CH3]4 (7), and [W6Cl8((mu-NC)Os(PPh3)2(C5H5))6][ p-OSO2C6H4CH3]4 (8). X-ray crystal structures are reported for 1, 4, and 5.     

Reignition phenomena after high-frequency current interruption withshort vacuum gaps

Based on a large number of measurements of high-frequency (HF) current interruptions in vacuum at small contact gaps (⩽600 μm), the statistical reignition behavior of vacuum switching devices after a HF current zero is investigated. Three types of reignitions can be classified. Statistical evaluation of post-arc current measurements for different parameters at current zero and different HF current ignition processes gives information about the stress of the gap as a result of the transient recovery voltage after a HF current zero (HF-TRV) and the accumulated post-arc charge. Comparing post-arc current values at the beginning of the HF-TRV and at the moment of reignition reveals a production of charge carriers during the recovery interval. Possible reasons for the different types of reignitions are discussed     

Radiofrequency spark source mass spectrometric analysis of oxygen in undoped silicon crystals and of carbon in undoped and carbon doped gallium arsenide crystals

Radiofrequency spark source mass spectrometry is a reliable and precise analytical method to measure the amount of oxygen in silicon grown by the Czochralski technique from SiO₂ crucibles in the common range from 2ppm(atomic) to 20ppm(atomic) and in silicon grown by the floating zone technique below 0.1ppm(atomic). The technique is also excellent for the measurement of the amount of carbon in semi-insulating gallium arsenide grown under low and high pressure N₂ ambient gas by the B₂O₃ encapsulated Czochralski technique from pyrolytic BN crucibles in the common range from 0.02ppm(atomic) to 0.4ppm(atomic). The results are in rather good agreement with concentrations measured by charged particle activation analysis and consistent with those obtained using other methods.