Roman Andrushkiw

A trace identity and its application to integrable systems of 1 + 2 dimensions

A trace identity is proposed and it is shown that this identity can be effectively applied to establish the Hamiltonian structure of the KP, DS hierarchies and a new hierarchy of (1+2)‐dimensional systems.

Mathematical modeling of freezing front propagation in biological tissue

A transient heat conduction problem involving contact freezing of biological tissue is investigated. The problem is modeled by a free boundary problem of the Stefan-type with phase-change and body heating terms. The effects of metabolic heat generation and blood perfusion in the tissue are taken into account in the model. The model is solved numerically by an implicit finite-difference scheme employing the enthalpy method. The numerical results predict the temperature profile in the tissue surrounding the cryogenic probe and the location of the freezing front boundary with respect to time.

Algebraic structure of the gradient‐holonomic algorithm for Lax integrable nonlinear dynamical systems. I

The algebraic structure of the gradient‐holonomic algorithm for Lax integrable dynamical systems is discussed. A generalization of the R‐structure approach for the case of operator‐valued affine Lie algebras is used to prove the bi‐Hamiltonian formulation of nonlinear integrable dynamical systems in multidimensions. The monodromy transfer matrix is constructed to describe the operator manifold in relation to canonical Lie–Poisson bracket on initial affine Lie algebra with gauge central extension. As an illustration, the two‐dimensional operator Benney–Kaup integrable hierarchy is considered and their bi‐Hamiltonicity is proved.

Geometric quantization of Neumann‐type completely integrable Hamiltonian systems

The procedure of geometric quantization is developed for completely integrable dynamical systems on manifolds with exact symplectic structure. These results are applied to Neumann’s nonharmonic oscillatory system on two‐dimensional sphere S2.

Algebraic structure of the gradient‐holonomic algorithm for Lax integrable nonlinear dynamical systems. II. The reduction via Dirac and canonical quantization procedure

The generalized theory of the R‐structure on affine operator Lie algebras is used to construct a complete theory of Lax integrable nonlinear dynamical systems in multidimensions. The operator bi‐Hamiltonian structures and their functional reductions are discussed in great detail in the examples of operator Korteweg–de Vries and Benney–Kaup dynamical systems. As an important by‐product of the developed algebraic theory, the Dirac canonical quantization problem is solved almost completely for the Neumann–Bogoliubov‐type oscillatory dynamical system on spheres, associated via Moser with the spectral moment map on an affine associative metrized Lie coalgebra with a one‐parameter gauge two‐cocycle. Some remarks are given on the problem of extending the developed algebraic theory to the case of Lax integrable dynamical systems on discrete manifolds.

Analysis of Malignancy‐Associated DNA Changes in the Nuclei of Buccal Epithelium in the Pathology of the Thyroid and Mammary Glands

Abstract: The object of the investigation reported in this paper was to study, from the point of view of statistical and geometric theory of pattern recognition, the DNA optical density distribution peculiarities in the interphase nuclei of buccal epithelium present in the pathology of the thyroid and mammary glands. Two new indices to characterize this distribution (ratio of modal class volumes and relief index) are proposed. It is shown that in malignant neoplasms of the thyroid and mammary glands the changes in the nuclei of buccal epithelium are characterized by an increase in the optical density of DNA over a range from 0.15 to 0.30 in conventional units of measure, as compared with its values in benign pathological processes. The sensitivity of the proposed criterion for diseases of the thyroid gland is equal to 76.2% and the specificity is equal to 85.8%. For diseases of the mammary gland (excluding IDLC) we have discovered that the sensitivity of the method is equal to 94.29% and its specificity equal to 90.91%. In diseases of the mammary gland (including IDLC) we have discovered that the sensitivity of the method is equal to 71.42% and its specificity is equal to 90.91%.

Interface dynamics for quasi-stationary Stefan problem

We investigated the interface dynamics in a Laplacian growth model, using the conformal mapping technique. Starting from the governing equation obtained by B. Shraiman and D. Bensimon, we derive intergro-differential evolution equation of interphase dynamics. It is shown that such representation of the conformal mapping technique is convenient for computer simulations of the quasi-stationary Stefan problem.

Iterative method for a class of nonlinear eigenvalue problems

Let H Be a complex and separable Hilbert space and consider in H the nonlinear eigenvalue problem where A, B, and C belong to the class of unbounded nonsymmetric operators, which are K- positive K-symmetric. Sufficient conditions insuring the existence of the eigenvalues of (i) are investigated. An iterative method for approximating the eigenvalues of (i) is developed and its convergence proved. Some numerical examples are given to illustrate the theory.