Zolotarev problem in the metric of L1([−1, 1])

In this paper we solve the problem of the determination of a polynomial of degree n with given two leading coefficients which has the least deviation from zero in the metric of L₁ ([?1, 1]). The extremal polynomial is expressed in the form of some linear combination of Chebyshev polynomials of the second kind.     

On the Dynamics of Space Debris: 1:1 and 2:1 Resonances

We study the dynamics of the space debris in the 1:1 and 2:1 resonances, where geosynchronous and GPS satellites are located. By using Hamiltonian formalism, we consider a model including the geopotential contribution for which we compute the secular and resonant expansions of the Hamiltonian. Within such model we are able to detect the equilibria and to study the main features of the resonances in a very effective way. In particular, we analyze the regular and chaotic behavior of the 1:1 and 2:1 resonant regions by analytical methods and by computing the Fast Lyapunov Indicators, which provide a cartography of the resonances. This approach allows us to detect easily the location of the equilibria, the amplitudes of the libration islands and the main dynamical stability features of the resonances, thus providing an overview of the 1:1 and 2:1 resonant domains under the effect of Earth’s oblateness. The results are validated by a comparison with a model developed in Cartesian coordinates, including the geopotential, the gravitational attraction of Sun and Moon and the solar radiation pressure.     

关于函数方程f_1~(n_1)+af_1~(m_1)f_2~(m_2)+f_2~(n_2)=1

对于函数方程f1n1+af1m1f2m2+f2n2=1,其中a∈C/{0},n1,n2,m1,m2∈N,给出存在非常数亚纯函数解和整函数解的必要条件.     

Fundamental theorem of geometry without the 1-to-1 assumption

It is proved that any mapping of an -dimensional affine space over a division ring onto itself which maps every line into a line is semi-affine, if and . This result seems to be new even for the real affine spaces. Some further generalizations are also given. The paper is self-contained, modulo some basic terms and elementary facts concerning linear spaces and also - if the reader is interested in other than , , or - division rings.

     

On the integrability of the (1+1)-dimensional and (2+1)-dimensional Ito equations

Under investigation in this paper are the (1+1)-dimensional and (2+1)-dimensional Ito equations. With the help of the Bell polynomials method, Hirota bilinear method and symbolic computation, the bilinear representations, N-soliton solutions, bilinear Bäcklund transformations and Lax pairs of these two equations are obtained, respectively. In particular, we obtain a new bilinear form and N-soliton solutions of the (2+1)-dimensional Ito equation. The bilinear Bäcklund transformation and Lax pair of the (2+1)-dimensional Ito equation are also obtained for the first time. Copyright © 2014 John Wiley & Sons, Ltd.     

On the bases of the spaces L1[0, 1] and C[0, 1]

New and simple proofs are given for the non-existence of unconditional bases in the spaces L₁[0, 1] and C[0, 1].Translated from Matematicheskie Zametki, Vol. 11, No. 3, pp. 241–249, March, 1972.The author wishes to thank P. L. Ul'yanov for his interest in the problem under consideration.