For an arithmetic semigroup (*G*, *∂*), we define entropy as a function on a naturally defined continuous semigroup *Ĝ* containing *G*. The construction is based on conditional maximization, which permits us to introduce the conjugate variables and the Lagrangian manifold corresponding to the semigroup (*G*, *∂*).

For an algebra A, denote by VA(n) the dimension of the vector space spanned by the monomials whose length does not exceed n. Let TA(n) = VA(n) − VA(n − 1). An algebra is said to be boundary if TA(n) − n < const. In the paper, the normal bases are described for algebras of slow growth or for boundary algebras. Let L be a factor language over a finite alphabet A. The growth function TL(n) is the number of subwords of length n in L. We also describe the factor languages such that TL(n) ≤ n + const.

We consider the problem of a viscous incompressible fluid flow along a flat plate with a small solitary perturbation (of hump, step, or corner type) for large Reynolds numbers. We obtain an asymptotic solution in which the boundary layer has a double-deck structure.

A relation between the jump of spin and the corresponding jump of energy is derived. This relation is used to determine the binding energy of the nucleus and the “entanglement” energy between two bosons. The latter is shown to be inversely proportional to the area in the two-dimensional case.

After the work of Irving Fisher, who formulated the main law of economics as an analog of the Claiperon--Clausius--Mendeleev relation corresponding to the Boltzmann--Maxwell ideal gas, the authors have developed this approach on the basis of the general principles of UD-statistics by assigning to the nominal interest rate the role of the chemical potential mu (with a minus sign).

The problem of the ∗-weak decomposability into ergodic components of a topological N0-dynamical system (Ω, ϕ), where ϕ is a continuous endomorphism of a compact metric space Ω, is considered in terms of the associated enveloping semigroups. It is shown that, in the tame case (where the Ellis semigroup E(Ω, ϕ) consists of endomorphisms of Ω of the first Baire class), such a decomposition exists for an appropriately chosen generalized sequential averaging method. A relationship between the statistical properties of (Ω, ϕ) and the mutual structure of minimal sets and ergodic measures is discussed.

The problem of the ∗-weak decomposability into ergodic components of a topological N0-dynamical system (Ω, ϕ), where ϕ is a continuous endomorphism of a compact metric space Ω, is considered in terms of the associated enveloping semigroups. It is shown that, in the tame case (where the Ellis semigroup E(Ω, ϕ) consists of endomorphisms of Ω of the first Baire class), such a decomposition exists for an appropriately chosen generalized sequential averaging method. A relationship between the statistical properties of (Ω, ϕ) and the mutual structure of minimal sets and ergodic measures is discussed.

We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.

We study the complexity of the realization of Boolean functions by circuits in infinite complete bases containing all monotone functions with zero weight (cost of use) and finitely many nonmonotone functions with unit weight. The complexity of the realization of Boolean functions in the case where the only nonmonotone element of the basis is negation was completely described by A. A. Markov: the minimum number of negations sufficient for the realization of an arbitrary Boolean function *f* (the inversion complexity of the function *f*) is equal to ]log_2(*d*(*f*) + 1)[, where *d*(*f*) is the maximum (over all increasing chains of sets of values of the variables) number of changes of the function value from 1 to 0. In the present paper, this result is generalized to the case of the computation of Boolean functions over an arbitrary basis *B* of prescribed form. It is shown that the minimum number of nonmonotone functions sufficient for computing an arbitrary Boolean functionf is equal to ]log_2(*d*(*f*)/*D*(*B*)+1)[, where *D*(*B*) = max *d*(ω); the maximum is taken over all nonmonotone functions ω of the basis *B*.

Noncooperative discounted stochastic n-person games are considered; the payoffs at each step are represented by trapezoidal fuzzy numbers. The existence of stationary Nash equilibrium strategies is proved.

A fluid flow along a semi-infinite plate with small periodic irregularities on the surface is considered for large Reynolds numbers. The boundary layer has a double-deck structure: a thin boundary layer (“lower deck”) and a classical Prandtl boundary layer (“upper deck”). The aim of this paper is to prove the existence and uniqueness of the stationary solution of a Rayleigh-type equation, which describes oscillations of the vertical velocity component in the classical boundary layer.

In this paper, 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 prove that the problem of finitely smooth equivalence can be solved for such systems by using finite segments of the Taylor series of their right-hand sides.

We try to explain the physical meaning of the notion of liquid “without interaction” and its characteristic property of having a small number of degrees of freedom. We show the relationship between opalescence and turbulence.

As is well known, the two-parameter Todd genus and the elliptic functions of level d define n-multiplicative Hirzebruch genera if d divides n + 1. Both cases are special cases of the Krichever genera defined by the Baker–Akhiezer function. In the present paper, the inverse problem is solved. Namely, it is proved that only these properties define n-multiplicative Hirzebruch genera among all Krichever genera for all n.

A mechanism for the inheritance of properties of spectra by differential spectra is developed and applied to prove geometric properties of morphisms of differential algebraic varieties.

We discuss the general opportunity to create (asymptotically) a comletely integrable system from the original perturbed system by inserting additional perturbing terms. After such an artificial insertion, there appears an opportunity to make the secondary averaging and secondary reduction of the original system. Thus, by this way, the $3D$-system becomes $1$-dimensional. We demonstrate this approach by the example of a resonance Penning trap.