Scientific heritage of Boris Lukich Laptev (4/23/05–1/15/89)

Biographical information on the eminent Soviet geometer Boris Lukich Laptev (4/23/05–1/15-89) is given as well as a survey of his research in the areas of differential geometry and history of mathematics.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 21, pp.27–42, 1989.     

A Second-Order Theory for Electron Plasma Solitary Waves in a Cylindrical Waveguide

To describe the long time asymptotic behaviour of weakly nonlinear plasma waves propagating in a strongly magnetized plasmafilled cylindrical waveguide the usual KDV equation which is first order in wave amplitude is extended to second order including the effects of finite temperature, mobile ions and giving a proper treatment to the radial co-ordinate for all possible modes of propagation. The second-order effects on the first-order solitary wave profile, phase speed and first order correction to the wavelength are all determined and discussed.     

Adaptive local grid refinement algorithms for finite-element collocation

An adaptive grid refinement procedure allows accurate solutions to advection-dominated, time-dependent flows using finite-element collocation. The technique relies on a data structure that is readily amenable to parallel computing. The paper discusses computational aspects of the method.     

On Critical Stability of Three Quantum Charges Interacting Through Delta Potentials

We consider three one-dimensional quantum, charged and spinless particles interacting through delta potentials. We derive sufficient conditions which guarantee the existence of at least one bound state.     

Asymptotic integration of a problem of convection

Rostov-on-Don. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 30, No. 4, pp. 69–75, July–August, 1989.     

Butyltins in california river and lake marina waters

Butyltins were analyzed in waters from California river and lake marinas and were detected at the part per trillion (ppt) level in most locations. The tributyltin:dibutyltin (TBT/DBT) ratios in the fresh waters were similar to those in their saline counterparts indicating like rates of degradation in the two salinity regimes.     

Discontinuous travelling wave solutions for certain hyperbolic systems

In an earlier paper on a malignant cell invasion model (Marchantet al., SIAM J. Appl. Math, 60, 2000) we introduced a novelform of discontinuous travelling wave solution. These solutionscould be studied easily by combining behaviour within a phaseplane with the Rankine–Hugoniot shock conditions, whichdescribe properties (such as the ratio of the jump discontinuitiesto the speed of propagation) that solutions may possess. Theseresults were new for several reasons. The shock conditions relateto hyperbolic equations (which the model is) but were appliedin a travelling wave ordinary differential equation phase planeusing techniques that usually apply to parabolic reaction–diffusionsystems. In addition the solutions possess singular behaviournear several points in the phase plane but in spite of thisthere exists a robust and stable family of physically interestingsolutions. In this paper we discuss two previously studied models, oneof detonation theory and one of angiogenesis. We show that eachof these models also possesses a family of discontinuous travellingwave solutions which was not previously discovered. Of particularinterest is the solution which has a blunt interface at thefront of the invading profile. In all three models it is thissolution that is seen to stably evolve from physically relevantinitial data, and for physically relevant parameter values. This work confirms the robustness of these novel travellingwave solutions and their applicability to a wider range of mathematicalmodelling situations.     

Frequency-domain design of optimal controllers

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 5, pp. 615–621, May, 1989.     

Reductions in overshoot following intense sound exposures

Overshoot refers to the poorer detectability of brief signals presented soon after the onset of a masking noise compared to those presented after longer delays. In the present experiment, brief tonal signals were presented 2 or 190 ms following the onset of a broadband masker that was 200 ms in duration. These two conditions of signal delay were tested before and after a series of exposures to a tone intense enough to induce temporary threshold shift (TTS). The magnitude of the overshoot was reduced after the exposure when a TTS of at least 10 dB was induced, but not when smaller amounts of TTS were induced. The reduction in overshoot was due to a decrease in the masked thresholds with the 2-ms delay; masked thresholds with the 190-ms delay were not different pre- and post-exposure. The implication is that the mechanisms responsible for the normal overshoot effect are temporarily inactivated by the same stimulus manipulations that produce a mild exposure-induced hearing loss. Thus the result is the paradox that exposure to intense sounds can produce a loss of signal detectability in certain stimulus conditions and a simultaneous improvement in detectability in other stimulus conditions.