The complete set of symmetry operators of the schrödinger equation

The complete set of symmetry operators of an arbitrary order associated with the Schrödinger equation is found. It is shown that this equation is invariant with respect to a 28-dimensional Lie algebra, realized in the class of differential operators of the second order. Higher-order symmetries of the Levi-Leblond equation are investigated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 11, pp. 1521–1526, November, 1991.     

Reactions of the nitrate radical with a series of reduced organic sulphur compounds in air

A 480 L evacuable reaction chamber, equipped with FT-IR spectroscopy on-line and ion chromatography off-line, has been used to study the gas phase reaction between the nitrate radical, NO₃, and the reduced organic sulphur compounds CH₃CH₂SH, (CH₃CH₂)₂S, (CH₃CH₂)₂S₂, and CH₃CH₂SCH₃ in air. The products CH₃CH₂SO₃H, SO₂, H₂SO₄, CH₃CHO, and CH₃CH₂ONO₂ were identified and quantified in the reactions of the first three compounds, CH₃CH₂SH, (CH₃CH₂)₂S, and (CH₃CH₂)₂S₂. The reaction products were CH₃CH₂SO₃H, CH₃SO₃H, SO₂, H₂SO₄, CH₃CHO, and CH₂O in the reaction of CH₃CH₂SCH₃. On the basis of identified reaction products and intermediates observed in the infrared spectra, mechanisms are proposed for the reactions between the NO₃ radical and the four reduced organic sulphur compounds. The results of this study, together with those from previous experiments performed in this laboratory on CH₃SCH₃, CH₃SH, and CH₃SSCH₃ lead to the conclusion that all these species, in the reaction with the NO₃ radical, follow a similar degradation mechanism producing SO₂, H₂SO₄, R? SO₃H, R? CHO, and R? CH₂ONO₂, as the main reaction products. The inital step of the reaction of NO₃ with R? S? R and R? S? H type (R = CH₃, CH₂CH₃) reduced organic sulphur compounds was found to be H-atom abstraction, probably after the formation of an initial adduct. For the reaction between NO₃ and R? S? S? R type compounds, evidence for an addition-decomposition reaction, as the initial steps, was obtained. R? S·, R? S(O)·, and R? S(O)₂· appear to be formed as intermediates in all the reactions. © John Wiley & Sons, Inc.     

Analysis of fragrance compounds in blood samples of mice by gas chromatography, mass spectrometry, GC/FTIR and GC/AES after inhalation of sandalwood oil.

After inhalation experiments with sandalwood oil and the pure fragrance compounds coumarin and alpha-terpineol, substances were detected and measured in the blood samples of test animals (mice) using gas chromatography/mass spectrometry (GC/MS) (MID) in connection with GC/FTIR (SWC), GC/AES (carbon and oxygen trace) and flame ionization detection/gas chromatography. Using tiglinic acid benzyl ester as the internal standard the following concentrations in serum could be found: alpha-santalol 6.1 ng/mL, beta-santalol 5.3 ng/mL and alpha-santalene 0.5 ng/mL. In separate inhalation experiments with coumarin and with alpha-terpineol the corresponding concentrations were 7.7 ng/mL and 6.9 ng/mL, respectively.     

Maximal triangle-free graphs with restrictions on the degrees

We investigate the problem that at least how many edges must a maximal triangle-free graph on n vertices have if the maximal valency is ≤D. Denote this minimum value by F(n, D). For large enough n, we determine the exact value of F(n, D) if D ≥ (n ? 2)/2 and we prove that lim F(n, cn)/n = K(c) exists for all 0 < c with the possible exception of a sequence c_k → 0. The determination of K(c) is a finite problem on all intervals [γ, ∞). For D = cn^?, 1/2 < ? < 1, we give upper and lower bounds for F(n, D) differing only in a constant factor. (Clearly, D < (n - 1)^1/2 is impossible in a maximal triangle-free graph.)     

Alfven's programme in solar system physics

Hannes Alfven's research program in solar system physics provides a valuable case study for the historians and philosophers of science. The response of the scientific community to his remarkable contributions is a severe test of widely held views about how science is or should be done. Seven of those contributions are examined, in the light of two doctrines of scientific methodology, those of Karl Popper (1959) and Imre Lakatos (1978). The seven are: application of the magnetic braking concept to the origin of the solar system; magnetohydrodynamic waves; field-aligned (`Birkeland') currents; critical ionization velocity and the existence of planetary rings; jet streams; electrostatic double layers; and partial corotation     

Theory of a frequency-step-tunable gyrotron for optimum plasma ECRH

A theory is presented which describes the dependence of the multimode operation of a gyrotron on the modulation and beam voltages. Nonlinear nonstationary processes are considered. An approximate start-up scenario of a gyrotron is determined which makes it possible to achieve single-mode excitation and high-efficiency operation in a desired mode, as well as to switch the tube to a similar operation in other modes     

Ungleichungen für elementare Mittelwerte

Ohne Zusammenfassung     

Using the method of integral relationships to determine the seepage parameters of a layer

We consider nonlinear seepage with an initial pressure gradient in a region without stagnant zones. An algorithm based on the method of integral relationships is proposed for the determination of an auxiliary function defined as the square root of the permeability coefficient of the layer. The advantage of this method is that it reduces the problem to a form which can be solved by standard computer subprograms. Applications of the proposed approach are described.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 55, pp. 87–92, 1985.