Emden–Fowler-type equations of arbitrary order are considered. Lower and sharp asymptotic estimates of the nonoscillating continuable solutions of these equations are established.

In a neighborhood of a singular point, we consider autonomous systems of ordinary differential equations such that the matrix of their linear part has one zero eigenvalue, while the other eigenvalues lie outside the imaginary axis. We study the reducibility of such systems to polynomial normal form.

The paper contains the results of the study of the asymptotic proprties of solutions with integer-valued asymptotics as wel as of solutions arising from the rapid decrease of the coefficient of the equation. To analize the asymptotic behavior of solutions of the equations, methods of power geometry are used.

The paper deals with solutions to Emden-Fowler-type equations of any arbitrary order. The asymptotic properties of solutions to these equations are studied and a systematic survey of numerous uncoordinated results of analysis of continuable and noncontinuable solutions is given.

A new upper bound for the additive energy of the Heilbronn subgroup is found. Several applications to the distribution of Fermat quotients are obtained.

Emden-Fowler type equations of arbitrary order are considered. The paper contains asymptotic estimates of nonoscillating continuable and noncontinuable solutions of such equations.

In this paper we consider the first boundary-value problem for elliptic systems of high order, defined on unbounded domains, which solutions satisfy a condition of finiteness of the Dirichlet integral, also known as the energy integral

The problem of hidden parameters arose from the very origins of quantum mechanics. It was felt that quantum mechanics does not describe all natural phenomena and there must exist certain hidden parameters that would help find a classical foundation of quantum theory and thereby fill its gaps. In paper are considered Hidden parameter, measurement time, Go¨del numeration, quantum particles, classical particles.

We consider various normalizations enabling us to change the scale of the graphs of isotherms and isochores. The relationship between parastatisticsal value of the maximal number of particles, corresponding to a given energy, and temperature allows us to pass in the domain of positive chemical potentials from the parastatisticsal number *K *(*K *= *∞ *corresponds to Bose statistics and *K *= 0 to Fermi statistics) to the temperature, which changes the scaling of the pressure in this domain.

For a system of identical Bose particles sitting at integer energy levels with the probabilities of microstates given by a multiplicative measure with ≥ 2 degrees of freedom, we estimate the probability of the sequence of occupation numbers to be close to the Bose–Einstein distribution as the total energy tends to infinity. We show that a convergence result earlier proved by A.M. Vershik [Functional Anal. Appl. 30 (2), 95–105 (1996)] is a corollary of our theorems.

Sufficient conditions for a generalized solenoid to be realized as a hyperbolic attractor of shere diffeomorphisms are obtained. The main theorem and its corollaries allow one to construct examples of attractors with various properties.

The results obtained in this paper are related to the Palis–Pugh problem on the existence of an arc withfinitely or countably many bifurcations which joins two Morse–Smale systems on a closed smooth manifoldMn . Newhouse and Peixoto showed that such an arc joining flows exists for any nand, moreover, it is simple. However, there exist isotopic diffeomorphisms which cannot be joined by a simple arc. Forn=1, this is related to the presence of the Poincar´ erotationnumber, and forn=2, to the possible existence of periodic points of different periods and heteroclinic orbits. In this paper, for the dimensionn=3, a new obstruction to the existence of a simple arc is revealed, which is related to the wild embedding of all separatrices of saddle points. Necessary and sufficient conditions for a Morse–Smale diffeomorphism on the 3-sphere without heteroclinic intersections to be joined by a simple arc with a“source-sink”diffeomorphism are also found.

Key words: hyperbolic attractor, generalized solenoid, braid group, mapping class group

It is shown that, between the values of the activity *a* = 1 and *a* < 1, there is a gap, which can be overcome by using additional energy. This energy is defined on the spinodal *a* = 1 (μ = 0) on the*P–Z* diagram and gives, in the parastatistical distribution, an additional term of Bose condensate type, which is also preserved for μ < 0. This term is the right-hand side of the Fermi–Dirac distribution. In this paper, it is also shown how to find the “liquid–amorphous body” binodal.

The problem of sums of products in F_p×F_p is considered. An estimate for sums of products improving Bourgain’s result of 2005 is obtained. This estimate is applied to the problem of estimating polynomial exponential sums over multiplicative subgroups in F_p.