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Extremal problems for typically real polynomials go back to a paper by W. W. Rogosinski and G. Szeg\H{o}, where a number of problems were posed, which were partially solved by using orthogonal polynomials. Since then, not too many new results on extremal properties of typically real polynomials have been obtained. Fundamental work in this direction is due to M.~Brandt, who found a novel way of solving extremal problems. In particular, he solved C. Michel's problem of estimating the modulus of a typically real polynomial of odd degree. On the other hand, D. K. Dimitrov showed the effectivity of Fej\'er's method for solving the Rogosinski--Szeg\H{o} problems. In this article, we completely solve Michel's problem by using Fej\'er's method., Comment: 15 pages, 2 figure

We study the transport of surface electrons on superfluid helium through a microchannel structure in which the charge flow splits into two branches, one flowing straight and one turned at 90 degrees. According to Ohm law, an equal number of charges should flow into each branch. However, when the electrons are dressed by surface excitations (ripplons) to form polaron-like particles with sufficiently large effective mass, all the charge follows the straight path due to momentum conservation. This surface-wave induced transport is analogous to the motion of electrons coupled to surface acoustic waves in semiconductor 2DEGs., Comment: 5 pages, 3 figures

D. Dimitrov has posed the problem of finding polynomials that set the sharpness of the Koebe Quarter Theorem for polynomials and asked whether Suffridge polynomials are optimal. We disprove Dimitrov's conjecture for polynomials of degree 3, 4, 5 and 6. For polynomials of degree 1 and 2 the conjecture is obviously true. On the way we introduce a new family of polynomials that allows us to state a conjecture about the value of the Koebe radius for polynomials of a specific degree.

In 2002 Dimitar Dimitrov posted the problem of finding the optimal polynomials that provide the sharpness of Koebe Quarter Theorem for polynomials and asked whether Suffridge polynomials are optimal ones. We disproved Dimitrov's conjecture for polynomials of degree 3,4,5 and 6. For polynomials of degree 1 and 2 the conjecture is valid.

We present transport measurements in a quasi-1D system of surface electrons on liquid helium confined in a 101-$\mu$m long and 5-$\mu$m wide microchannel where an electrostatic potential with periodicity of $1$-$\mu$m along the channel is introduced. In particular, we investigate the influence of such a potential on the nonlinear transport of quasi-1D Wigner Solid (WS) by varying the amplitude of the periodic potential in a wide range. At zero and small values of amplitude, quasi-1D WS in microchannel shows expected features such as the Bragg-Cherenkov scattering of ripplons and reentrant melting. As the amplitude of potential increases, the above features are strongly suppressed. This behavior suggests loss of the long-range positional order in the electron system, which is reminiscent of the re-entrant melting behaviour due to the lateral confinement of WS in the channel.

We present a new study of the nonlinear transport of a two-dimensional electron crystal on the surface of liquid helium confined in a 10 micrometer-wide channel in which the effective length of the crystal can be varied from 10 to 215 micrometers. At low driving voltages, the moving electron crystal is strongly coupled to deformation of the liquid surface arising from resonant excitation of surface capillary waves, ripplons, while at higher driving voltages the crystal decouples from the deformation. We find strong dependence of the decoupling threshold of the driving electric field acting on the electrons, on the size of the crystal. In particular, the threshold electric field significantly decreases when the length of the crystal becomes shorter than 25 micrometers. We explain this effect as arising from weakening of surface deformations due to radiative loss of resonantly-excited ripplons from an electron crystal of finite size, and we account for the observed effect using an instructive analytical model., Comment: 5 figures

For a pair of conjugate trigonometrical polynomials $C (t) = \sum_ { j = 1 } ^N { { a_j}\cos jt }, S(t) = \sum_ { j = 1 } ^N { { a_j}\sin jt }$ with real coefficients and normalization ${a_1} = 1 $ we solve the extremal problem \[ \sup_ {a_2,...,a_N} \left ({ \min_t \left\{ {\Re \left ({ F\left ({ { e^ {it} } } \right) } \right): \Im \left ({ F\left ({ { e^ {it} } } \right) } \right) = 0 } \right\} } \right) = -\frac14 \sec ^2\frac\pi{N + 2}. \] We show that the solution is unique and is given by \[ a_j^ {(0)} = \frac {1} { { { U'_N}\left ({\cos \frac{\pi } { { N + 2 } } } \right) } } { U' _ { N - j + 1 } }\left ({\cos \frac{\pi } { { N + 2 } } } \right) { U_ { j - 1 } }\left ({\cos \frac{\pi } { { N + 2 } } } \right), \] where the $U_j(x)$ are the Chebyshev polynomials of the second kind, and the $U'_j(x)$ are their derivatives, $j = 1, \ldots, N.$ As a consequence, we obtain some theorems on covering of intervals by polynomial images of the unit disc. We formulate several conjectures on a number of extremal problems on classes of polynomials.

A study was conducted to determine how the Career Preparation Assessment portfolio (CPA), tailored to be used in local school reform efforts, was implemented in 10 schools in 1998-99. The research was organized and reported using these four clusters of research questions: tailoring the CPA, implementation, student performance and assessment, and whole-school change. The study found that: (1) the tailoring process enabled schools to customize the CPA to their own needs and increase ownership; (2) the tailored CPA can be implemented successfully in schools possessing certain structural characteristics generally associated with schools that are engaged in a whole-school reform; (3) reliable scoring of the CPA requires a commitment to conducting scoring with substantial training; (4) student performance on the tailored CPA was acceptable to implementing teachers; and (5) a strong majority of project teachers believe the tailored CPA is a valuable teaching tool that promotes changes in classroom practice and can support school-wide reform. (Appendices include both original and tailored CPA standards and rubrics, and student, teacher and scorer survey instruments.) (KC)

The quasi-one-dimensional system of surface electrons over superfluid helium with negative charge on substrate is investigated. The temperature dependence of conductivity is ladder-like which is observed at temperature lower 1.3 K. The parameters of ladder have been different in different experiments. The observed dependence isn't explained by classical models of carry with account to the usual scatters in channel. It is supposed the dependence have been connected with quantum character of electron transport in narrow channels.

The electron transport in inhomogeneous quasi-one-dimensional conducting channels on the liquid helium surface are studied in the temperature range 0.6-1.5 K. Inhomogeneities are created by charging the substrate on which the conducting channels are formed. It has been established that the electron conductivity practically does not depend on temperature at some substrate charge. The results obtained are explained by localization of carriers and creation of electron polarons., Comment: 6 pages, 3 figures