In this paper, an application of the Riquer-Thomas-Janet theory is described for the problem of transforming a system of partial differential equations into a passive form, i.e., to a special form which contains explicitly both the equations of the initial system and all their nontrivial differential consequences. This special representation of a system markedly facilitates the subsequent integration of the corresponding differential equations. In this paper, the modern approach to the indicated problem is presented. This is the approach adopted in the Knuth-Bendix procedure [13] for critical-pair/completion and then Buchberger's algorithm for completion of polynomial ideal bases [13] (or, alternatively, for the construction of Groebner bases for ideals in a differential operator ring [14]). The algorithm of reduction to the passive form for linear system of partial differential equations and its implementation in the algorithmic language REFAL, as well as the capabilities of the corresponding program, are outlined. Examples illustrating the power and efficiency of the system are presented. 相似文献
The article surveys the main results of the statistical approach to the solution of ill-posed problems of mathematical physics, in application to specific ill-posed inverse problems in geophysics.Invited paper presented at the International Seminar on Mathematical Foundations of the Interpretation of Geophysical Fields, Moscow, May–June 1972.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 79, pp. 67–81, 1978. 相似文献
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.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 320, 2004, pp. 69–79. 相似文献
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.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2005, pp. 15–23. 相似文献
A test for a function to be a solution of an elliptic PDE is given in terms of extensions, as solutions, from the boundaries inside the domains belonging to an isotopic family. It generalizes a result of Ehrenpreis for spheres moved along a straight line.
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.
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Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 25, pp. 249–294, 2005. 相似文献
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 Es 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 Es and weak Pc3 pulsation.