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.     

Structure of norbornene-fused 3,1-oxazine double cycloadducts

Pentacyclic isoxazolines were obtained by the cycloaddition of benzonitrile oxide to norbornene-azetidinone-fused 3,1-oxazines. The constitutions of two of the isomers obtained, and the configurations and conformations of all products, were determined by means of ¹H and ¹³C NMR spectroscopy and DNOE experiments.     

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.     

The study of lithium fast ion conductors of Li3xLa0.67−xScyTi1−2yNbyO3 system

Perovskite-type lithium fast ion conductors of Li_3xLa_0.67−xSc_yTi_1−2yNb_yO₃ system were prepared by solid state reaction. X-Ray powder diffraction shows that perovskite solid solution form in the ranges of x=0.10, y≤0.10. AC impedance measurements indicate that the bulk conductivities and the total conductivities are of the order of 10⁻⁴ S·cm⁻¹ and 10⁻⁵ S·cm⁻¹ at 25 °C respectively. The compositions have low bulk activation energies of about 17 kJ/mol in the temperature ranges of 298 – 523 K and total activation energies of about 37 kJ/mol in the temperature ranges of 298 – 523 K.     

Effect of end groups on complexation kinetics between cyclodextrins and guest polymers

α‐Cyclodextrin (α‐CD) has been complexed with various poly(ethylene glycol) (PEG) derivatives in aqueous solution. It has been found that the end groups of PEG derivatives affect the complexation kinetics greatly, but have only a little influence on the thermodynamic behavior. By increasing the hydrophobicity of end groups, the complexation speeds up rapidly. On the other hand, the bulky end groups slow down the threading of polymeric guests into the cavity of CD. By changing the hydrophobicity and the size of end groups, the complexation rate can be adjusted in the range of several orders of magnitudes, which should be quite useful in the design of new supramolecular systems. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 2050–2057, 2006     

A mixed‐FEM formulation for nonlinear incompressible elasticity in the plane

This article deals with an expanded mixed finite element formulation, based on the Hu‐Washizu principle, for a nonlinear incompressible material in the plane. We follow our related previous works and introduce both the stress and the strain tensors as further unknowns, which yields a two‐fold saddle point operator equation as the corresponding variational formulation. A slight generalization of the classical Babu?ka‐Brezzi's theory is applied to prove unique solvability of the continuous and discrete formulations, and to derive the corresponding a priori error analysis. An extension of the well‐known PEERS space is used to define an stable associated Galerkin scheme. Finally, we provide an a posteriori error analysis based on the classical Bank‐Weiser approach. © 2002 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 18: 105–128, 2002     

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.     

Fast and Slow Diffusion of Channelized Particles in Transverse Energy Space

The Fokker–Planck-type kinetic equation is constructed with the help of the evolution equation for the transverse energy of channelized particles, which in its turn is derived beyond the framework of perturbation theory proceeding from the condition of nonconcervation of the adiabatic invariant.