On the distribution of the number of real zeros of a random polynomial

In this note, we find the distibution of the number of real zeros of a random polynomial. We also derive a formula for the expected number of complex zeros lying in a given domain of the complex plane. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 320, 2004, pp. 69–79.     

The matrix equation AX-YB=C and related problems

The main result of the paper is a theorem, using which a new proof of Roth’s theorem is obtained, a new solvability criterion for the matrix equation AX-YB = C is proved, a formula for a particular solution of the latter is derived, and the least of the orders of square nonsingular matrices containing a given rectangular matrix as a submatrix is determined. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2005, pp. 15–23.     

Definition and properties of characteristic frequencies of a linear equation

The definition of the characteristic frequencies of zeroes and changes of sign for solutions is given. It is equal to the upper medium (with respect to the time half-axis) of their number on the half-interval of length π. We also define the main frequencies for a linear homogeneous equation of order n. These main frequencies for an equation with constant coefficients coincide with the absolute values of the imaginary parts of the roots of the corresponding characteristic polynomial. It is proved that for the second-order equation the main frequencies are the same for all solutions and that they are stable with respect to uniformly small and infinitely small perturbations of the coefficients. For the third-order equation they can be different, and for any of the main frequencies an example of nonstability is given. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 25, pp. 249–294, 2005.     

Electrochemical Recovery of Silver from Lead Alloys in Chloride Melts

Electrochemical separation of lead-silver alloys into the constituents by thin-layer electrolysis in molten electrolytes, with bismuth as metal separator, is studied. It is proposed to determine the extent of component recovery by potentiometry.     

The polar ionosphere at Zhongshan station on May 11, 1999, the day the solar wind almost disappeared

The solar wind almost disappeared on May 11, 1999: the solar wind plasma density and dynamic pressure were less than 1cm⁻³ and 0.1 nPa respectively, while the interplanetary magnetic field was northward. The polar ionospheric data observed by the multi-instruments at Zhongshan Station in Antarctica on such special event day was compared with those of the control day (May 14). It was shown that geomagnetic activity was very quiet on May 11 at Zhongshan. The magnetic pulsation, which usually occurred at about magnetic noon, did not appear. The ionosphere was steady and stratified, and the F₂ layer spread very little. The critical frequency of day-side F₂ layer, f₀F₂, was larger than that of control day, and the peak of f₀F₂ appeared 2 hours earlier. The ionospheric drift velocity was less than usual. There were intensive auroral E_s appearing at magnetic noon. All this indicates that the polar ionosphere was extremely quiet and geomagnetic field was much more dipolar on May 11. There were some signatures of auroral substorm before midnight, such as the negative deviation of the geomagnetic H component, accompanied with auroral E_s and weak Pc3 pulsation.

     

Quadratic data envelopment analysis

Data Envelopment Analysis (DEA) offers a piece-wise linear approximation of the production frontier. The approximation tends to be poor if the true frontier is not concave, eg in case of economies of scale or of specialisation. To improve the flexibility of the DEA frontier and to gain in empirical fit, we propose to extend DEA towards a more general piece-wise quadratic approximation, called Quadratic Data Envelopment Analysis (QDEA). We show that QDEA gives statistically consistent estimates for all production frontiers with bounded Hessian eigenvalues. Our Monte-Carlo simulations suggest that QDEA can substantially improve efficiency estimation in finite samples relative to standard DEA models.     

VIBRATION OF TWISTED AND CURVED CYLINDRICAL PANELS WITH VARIABLE THICKNESS

A numerical procedure, with an exact strain-displacement relationship of twisted and curved cylindrical panels having variable thickness derived by considering the Green strain tensor on general shell theory, is presented using the principle of virtual work and the Rayleigh-Ritz method with algebraic polynomials as in-plane and transverse displacement functions. The accuracy and applicability of the procedure are verified by comparing the present results with previous experimental and theoretical results for several panels. The effects of variation ratio of thickness in chordwise and lengthwise directions, twist, and curvature both in two directions aforementioned on vibrations of cylindrical panels are studied in detail, and typical vibration mode shapes are plotted to demonstrate the effects.