A Miura of the Painlevé I Equation and Its Discrete Analogs

We present Miura transformations for the continuous and several discrete Painlev\'e I equations. In the case of the continuous P_I, we use the Hamiltonian formulation of the Painlev\'e equations and show that there exists a Miura transformation between P_I and the binomial, second degree, equation of Cosgrove SD_V. In the case of the discrete P_I's we obtain two different kinds of Miuras. One kind relates a d-P_I to some other d-P_I while the other leads to discrete four-point equations which are the discrete analogs of the derivative of Cosgrove's equation SD_V.     

Numerical modeling of a XeCl laser excited by microwave discharge

12 , 169 (1987). Reasonably good agreements in the peak output power and laser efficiency have been achieved. Model calculations also predict that an efficiency as high as 2.7% can be obtained once the conditions of the above-mentioned experiments have been optimized. From the consideration that the skin depth effectively limits the absorption length of the microwave pumping and hence the excitable volume, it is concluded that high input power densities (>2 MW/cm³) and higher gas pressures (between 3 and 10 atm) are the preferable conditions to achieve higher efficiency. Preliminary calculations on CCl₄ containing XeCl gas mixtures show that improvement in laser efficiency by several folds may be achieved as a result of the higher intrinsic efficiency of excimer formation. Received: 23 September 1996 / Revised version: 25 March 1997     

Quenching of fluorescence from Na(32P) and K(42P) atoms following photodissociation of NaCl and KCl at 193 nm

 Quenching of fluorescence from Na(3² P) and K(4² P) atoms by various collision partners was studied at 973 and 1273K. Excited alkali atoms were produced photolytically by excimer laser light at 193nm. For each collision pair, the appropriate relative velocity was computed and used to evaluate the quenching cross-section from the measured rate constants. Cross sections for CO₂, O₂ and N₂ are large (10–60Ų) while for Ar, the values are <1 Ų. The results are compared with those of previous investigations as a function of relative velocity. Finally, implications for combustion diagnostics are briefly discussed. Received: 29 March 1996     

Determination of the largest lyapunov index for short chaotic time sequences in self-oscillating chemical reactions

A method was developed for the determination of largest Lyapunov index for short chaotic time sequences with allowance for the presence of noise in the response of the system. For this purpose a simple unidimensional representation modelling the dynamics of irregular self-oscillations was investigated. Conditions were found for adequate determination of the largest Lyapunov index from the noise-distorted signals. The results were used to obtain the largest Lyapunov index for the Belousov—Zhabotinskii reaction taking place in an ideal-mixing closed reactor and in a constant-mixing flow-type reactor. Institute of Organic Chemistry, National Academy of Sciences of Ukraine. Translated from Teoreticheskaya i éksperimental’naya Khimiya, Vol. 33, No. 3, pp. 136–142, May–June, 1997.     

Influence of the domain structure on the permeability of planar hexaferrites in the natural ferrimagnetic resonance region

A calculation is presented of the permeability of hexaferrites with an easy magnetization plane in the region of natural ferrimagnetic resonance and in the presence of a domain structure. It is shown that the domain structure strongly influences both the magnitude of the permeability and the natural ferrimagnetic resonance frequency. The expressions obtained are compared with the results of calculations of the natural ferrimagnetic resonance spectra of single-domain planar hexaferrites. State University, Tomsk. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 53–57, February, 1997.     

Criteria for controllability and observability in complex domain models of stationary systems

We propose criteria for controllability and observability of stationary systems specified by complex domain models. Conclusions concerning controllability and observability are made on the basis of coefficients in the corresponding transfer functions. Translated fromDinamicheskie Sistemy, Vol. 12, pp. 120–126, 1993.     

Perfect Modules

Erika Mares introduced the concepts of semi-perfectness and perfectness for projective modules, generalised the structure theorems of H. Bass and obtained results on the endomorphism rings of such modules. The present author has carried out an extensive study of endomorphism rings of various types of modules with two of his collaborators [3], [9]. In particular the concept of semi-perfectness was extended to modules not necessarily projective and results similar to those of Erika Mares obtained for quasi-projective semi-perfect modules. The object of the present paper is to extend the concept of perfectness to modules which are not necessarily projective and obtain results similar to those of Erika Mares, Roger Ware etc. concerning these modules.     

Asymptotic behavior of neutron small-angle multiple scattering spectra

The asymptotic behavior of the spectra for large values of the scattering vector for the case of elastic multiple small-angle neutron scattering (SANS) is investigated theoretically and experimentally. An expansion of the spectrum in terms of the reciprocal of the magnitude of the momentum transfer is obtained taking account of the influence of the instrumental line. It is shown that, to within some factor, the leading term of the expansion is identical to the differential single-scattering cross section averaged over a statistical ensemble of particles; several subsequent terms in the expansion are calculated and the range of applicability of the resulting expressions is determined. The asymptotic behavior of the multiple SANS spectrum is measured, using a two-crystal neutron spectrometer, for samples of an HTSC ceramic, the alloy Fe-Ni, and Al powder. The agreement between the experimental results and the theoretical predictions is analyzed. Zh. éksp. Teor. Fiz. 114, 2194–2203 (December 1998)     

Duality in two-dimensional Z(N)-symmetric spin models on a finite lattice

We propose a method for deriving duality relations for two-dimensional inhomogeneous Z(N)-symmetric models on a finite square lattice wound around a torus. The method is used to obtain duality relations for the vector Potts model, the Berezinskii-Villain Z(N)-model, the Ashkin-Teller model, and the 8-vertex model on a lattice obliquely wound around a torus, as well as an exact relation linking the partition functions of the latter two models. Zh. éksp. Teor. Fiz. 113, 240–260 (January 1998)