Abbreviations used: P = Paper, C = Conference proceedings, R = Report. Numbers that follow refer to the list of my publications.
Modeling of reaction-diffusion and electrochemical processes.
Modeling gas pipeline corrosion.
Participation in studies related to the concept of the permanent underground
disposal of nuclear waste (R6):
- Corrosion modeling (C13-16,18,20,21,23,24,26,
R7,8,11,12,16,20-25,27-29,
P43) -
reaction-diffusion, electrochemistry; finite differences.
- Modeling other reaction-diffusion and diffusion-convection processes
using finite elements (C10,
R3,9,10,17), and other methods (C9,12,
R2).
- Analysing oxidation and dissolution experiments (P41, C19,22,25, R2).
- Surface, island-like oxidation (P40, P42).
- Helping with processing experimental data (P41, C11,17, R4),
mathematical consulting.
- Other studies (R15,18,19).
Highlights: In the corrosion modeling: treatment of very complex electrochemical boundary conditions and of precipitate or moisture layers of variable thickness; Method of moments applied to a moving-boundary diffusion-reaction problem (R2); Island formation on solid surfaces (P43).
Highlights: Some features of the TRANSIENT; Fast nonrecursive algorithms for the Tower of Hanoi puzzle.
- Quantum path-integral Monte-Carlo simulations: application to a
1-dimensional (1D) model of surface adsorption; formulation of an
efficient high-temperature approximation of the method (P39).
- Quasirandom number generation: found new fast algorithm for the
generation of the Halton numbers (P34).
- Study of 1D aperiodic deterministic systems (superlattices) -
continuation of the work started in Marseilles (P37,38, C8),
Thue-Morse system (P35).
- Investigation of dynamical systems - periodic windows in the Lorenz model
(C7), classical interacting electrons in magnetic fields
(P32).
Highlights: More extensive mapping of the Lorenz parameter space than ever before (C7); Testing random number generators by running my quantum MC code for the harmonic oscillator case (P39); Halton numbers (P33).
- Forefront study of the structure factor (diffraction spectra) of 1D aperiodic substitutional systems (superlattices), identification of new types of such diffraction spectra. From the mathematical point of view: study of a rather involved Fourier transform (of the density of scatterers of the probe radiation) (P36,37).
Highlight: Finding a new class of 1D quasicrystals (P36).
- Manipulation of multivariate polynomials - proof of a theorem on the minimum number of terms of two exactly divisible multivariate polynomials
that is sufficient for their recursive power series division (P33, C6).
- Participation in the testing and debugging of the new Japanese computer
algebra system GAL.
- Symbolic-numeric calculation of a class of definite integrals (C4).
- Investigation of classical chaos: study of the dynamics of a waterwheel which led
to finding the structure of the periodic windows for the Lorenz model and mapping the Lorenz parameter space (P29);
application of the dynamical systems theory to the study of a spin model (P31, C5).
- Study of n-letter substitutions as applied to the investigation of 1D nonperiodic
substitutional chains (classification, spectral and structural properties) (P26,27,29).
Highlights: Finding trace maps for arbitrary n-letter substitution rules (P27); Malkus waterwheel theory (P29).
- Nonperiodic ordered 1D structures (quasicrystals): Generalized Fibonacci
superlattices, calculation of the magnetic excitations in such structures
using periodic and real boundary conditions, investigation of the
properties of the dynamical trace maps associated with the generalized
Fibonacci sequences (pseudo-invariants, attractors), phase transition in
generalized Fibonacci quantum Ising models, properties of the generalized
Thue-Morse chains (P19-25).
- Semiclassical wave packet dynamics (scattering): Investigation of the
ways of minimizing the global error of the propagation methods in which
the wave-functions are approximated with Gaussian wave packets (P18, R1).
- Quantum chaos.
Highlights: Finding attractors of some of the Fibonacci trace maps (P21); Gaussian wave packet propagation (P18).
- Investigation of various mathematical and computational problems of the modeling of the evolution of biochemical systems, particularly the growth of giant single-celled algae: Types of area growth; discretized growth algorithms; formulation and "self-consistent" numerical solution (for the 1D and 2D cases) of coupled non-linear partial differential reaction-diffusion equations (e.g. of the Brusselator type) in a condensed medium that continuously expands everywhere in such a way that the local growth rate is proportional to one of the dependent variables (morphogen concentration) (P16).
Highlight: First self-consistent coupling of the growth with the reaction-diffusion equations for the 1D case (P16).
- Computation of electronic properties of liquid and amorphous metal clusters: clarification of the correct application of the recursion method in the case of non-orthogonal basis (overlapping atomic orbitals) - development of the recursion method for a general basis and a general GF element; testing of the approximations in the LCAO-type model used for these calculations (approximate Anderson pseudopotential, finite basis, exchange potential correction) (P14).
Highlight: Correct formulation of the recursion method for a non-orthogonal basis (P14).
- Field emission from the region of band gaps (P11,12,17).
- "Quantum-chemical" approach to transition metals (P13).
- Translating between various languages.
- Surface states. Magnetic surface anisotropy of nickel.
Integral characteristics of solid surfaces (P3-6,9,10).
- Local-density-functional atomic calculations (P8, C2).
- Calculation of electron structure of extended systems:
construction of ab-initio atomic pseudopotentials;
computer program for the self-consistent ab-initio
pseudopotential calculation of electronic structure of
semiconductors with zinc-blende lattice (intended for the
investigation of deep-level impurities) (P7, C3).
Highlights: Integral surface characteristics in the tight-binding formalism (P4); Construction of the atomic pseudopotentials (C3); Code for the calculation of the pseudopotential matrix elements (P7).
- Surface states and the magnetic surface anisotropy of nickel (P2, C1).
- Electron surface diffraction (P1).