On the Painlevé Property of Isomonodromic Deformations of Fuchsian Systems

We give a review of the modern theory of isomonodromic deformations of Fuchsian systems discussing both classical and modern results, such as a general form of the isomonodromic deformations of Fuchsian systems, their differences from the classical Schlesinger deformations, the Fuchsian system moduli space structure and the geometric meaning of new degrees of freedom appeared in a non-Schlesinger case. Using this we illustrate some general relations between such concepts as integrability, isomonodromy and Painlevé property. The work is supported by N.Sh.-6849.2006.1 and RFBR 07-01-00526 grants.     

Scientific contributions of Leo Khachiyan (a short overview)

On April 29, 2005, Leo Khachiyan passed away with tragic suddenness in the middle of his scientific career. He became famous for his polynomial time algorithm for linear programming (LP)[L. Khachiyan, A polynomial algorithm in linear programming, Soviet Math. Doklady 20 (1) (1979) 191-194; L. Khachiyan, Polynomial algorithms for linear programming, USSR Comp. Math. and Math. Phys. 20 (2) (1980) 51-68]. However, he contributed fundamentally to several other areas, as well. In this introductory paper, we would like to recall briefly his main contributions, and provide a complete (up to our best knowledge) list of his publications.     

Reflection Operator and Canonical Basis

We construct a canonical basis of the Hilbert space of automorphic functions such that the invariance group of this basis has a nontrivial multiplier. Bibliography: 7 titles.     

Novel types of boronated chlorin e6 conjugates via ‘click chemistry’

Conjugates of chlorin e₆ with closo‐dodecaborate and cobalt bis(dicarbollide) anions were synthesized for the first time in high yields using the advanced ‘click’ methodology. In vitro study on A549 human lung adenocarcinoma cells revealed that the synthesized boronated conjugates are able to penetrate and accumulate in cancer cells, but their intracellular concentration is not sufficient for effective photodynamic and boron neutron capture therapy of cancer. Copyright © 2009 John Wiley & Sons, Ltd.     

New phosphonium salts based on 3-(diphenylphosphino)propanoic and ω-haloalkanoic acids

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Heterodyne polarimetry for measuring critical dimensions on semiconductor wafers

A new optical technique based on heterodyne polarimetry is developed for fast measurement of critical dimensions on semiconductor wafers, particularly on those of memory chips. Sub-wavelength periodical structure of a sample acts as a wire-grid polarizer, making both the amplitude and phase of the reflected laser beam dependent on geometrical dimensions and optical properties of the wafer pattern. The heterodyne technology based on the cross-polarized two-frequency Zeeman laser is used to simultaneously measure the amplitude and the phase of the reflected laser beam. Very fast and efficient effective medium trapezoidal algorithm is developed for extracting values of critical dimension from the measurements. The technique is most suitable for inspection of multi-periodical patterns consisting of large number of periodically distributed small areas of sub-wavelength gratings. Theoretical background, simulation, and experimental results are presented.     

The boundary value problem and the Jensen inequality for an entropy functional of a Markov diffusion process

The paper introduces the recent results related to an entropy functional on trajectories of a controlled diffusion process, expressed through an additive functional of the diffusion process, with a Lagrangian, determined by the parameters of a controlled stochastic equation. These results include a minimum condition for the entropy functional and the functional's Jensen inequality, which both are useful for the solution of important mathematical and applied problems.     

Tridiagonal canonical matrices of bilinear or sesquilinear forms and of pairs of symmetric,skew-symmetric,or Hermitian forms

Tridiagonal canonical forms of square matrices under congruence or *congruence, pairs of symmetric or skew-symmetric matrices under congruence, and pairs of Hermitian matrices under *congruence are given over an algebraically closed field of characteristic different from 2.