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70th anniversary of the Great October Socialist dedicated to revolution

	Mathematics in the contemporary world V. A. Sadovnichii		3
Mathematics
	The complexity of monotone calculations of polynomials S. B. Gashkov		7
	The sum of a quadratic form and a power of an integer V. A. Plaksin		13
	The countable character of $X$ in $\beta X$ versus the countable character of the diagonal in $X\times X$ V. I. Ponomarev, V. V. Tkachuk		16
	Transfinite dimensions and mappings V. V. Fedorchuk		19
	Time change in integrable systems A. P. Veselov		25
	On some summation formulas G. I. Arkhipov, V. N. Chubarikov		29
	A characterization of Polish $AE(n)$-spaces A. Ch. Chigogidze		32
	The analyticity of sums of trigonometric series with infinite-monotone coefficients M. I. Dyachenko		35
	The properties of random trigonometric polynomials with $\pm1$ coefficients B. S. Kashin		40
Mechanics
	Dynamic regimes of multi-legged walking robots motion Yu. F. Golubev, E. V. Degtyareva, T. A. Tsar'kova		47
	Hypersonic flow at a rotating axially symmetriс body with massive blowing V. A. Levin, N. A. Krasilov		51
	Conditions of detonation stationarity in homogeneous medium V. M. Gendugov		55
	A qualitative investigation of an integrable variant of the nonholonomic Suslov problem G. G. Okuneva		59
	Expansion of the burned gas behind the detonation wave in a tube-formed explosive N. N. Smirnov		64
	On the problem of the periodic trajectories of the Birkhoff billiard D. V. Treschev		72
	The dynamics of systems with nonintegrable constraints. IV. Integral principles V. V. Kozlov		76
Short notes
	A constant in an embedding theorem Buĭ Ta Long		84
	Some properties of automorphic functions of rotation that are connected with the rate of growth of their spherical Ahlfors–Shimizu characteristic S. V. Kravtsev		87
	On a calculation method for turbulent near wall jets N. G. Vasetskaya, V. A. Ioselevich		90
	Momentum of forces acting on a rotating spherical particle in viscous non-isothermic gas A. F. Rogovoi		93
	Air-particle suspensions combustion in pipes in presence of friction and heat exchange P. B. Vainshtein, V. V. Popov		96
	The instability of equilibrium of a gyroscopic system with two degrees of freedom S. D. Furta		100
	To the eightieth birthday anniversary of P. M. Ogibalov		102