This paper is devoted to the formulation and proof of the theorems on the mean value of a polylinear function, similar to the direct and inverse theorems on the mean value of harmonic functions. It is proved that the value of an arbitrary polylinear function fP (x) at the central point of G—an arbitrary n-dimensional coordinate parallelepiped—is equal to the mean value of the function fP (x) over the set of k-dimensional faces G for any k \in \{ 0, . . . , n\} . Based on this, it is justified that just once, by calculating the value of the polylinear continuation fP (x) of an arbitrary Boolean function fB(x) at the central point of an n-dimensional unit cube, one can find the number of Boolean vectors on which the Boolean function fB(x) takes the value 1 and thereby, in particular, determine the satisfiability of the Boolean function fB(x). It has also been established that such a property is characteristic only of polylinear functions, i.e., it has been proven that if for any G — n-dimensional coordinate parallelepiped and at least for some number k \in \{ 0, . . . , n\} , the value of the continuous function f(x) at the central point G is equal to the mean value of the function f(x) over the set of k-dimensional faces of G, then the function f(x) is polylinear.

We consider the family of Schrödinger operators Hγλ(K), which are associated with the Hamiltonian of a system of two identical bosons on the d-dimensional lattice Zd, where d≥3, with interactions on each site and between nearest-neighbor sites with strengths γ∈R− and λ∈R−, respectively. Here, K∈Td is a fixed quasi-momentum of the particles. We first partition the (γ,λ)−plane into connected components S0, S1 and Cj,j=0,1,2. Further, we establish below-threshold effects for Hγλ(0) on the boundaries of the connected  components ∂S0 and ∂Cj,j=0,2.

T. Huber, D. Schultz and D. Ye have obtained four Eisenstein series identities of level 20 and employed them to derive a new series for 1 \pi . Motivated by this, in this article, nine new Eisenstein series identities of level 20 are presented and their combinatorial properties are given. 

The study of special series in Meixner polynomials, that was started in the author’s previous works, is continued. The present paper is devoted to the study of approximation properties of de la Vall´ee Poussin means for partial sums of the mentioned series. It is shown that for a function f the rate of the weighted approximation by Vall´ee Poussin means has the same order as the best weighted approximation of f. 

The article introduces a characteristic of a triangle, reflecting the measure of its nondegeneracy. The importance of studying this quantity is associated with the construction of highquality computational grids. It is shown that if a Sobolev class mapping distorts this characteristic multiple times, then this mapping is a mapping with limited distortion. In addition, it is proved that if the above condition and additionally the condition of limited distortion of the area of the triangle are satisfied, then the mapping is bi-Lipschitz. The article establishes estimates for all constants characterizing the mappings under study. 

A rational function of two complex variables and all its Laurent expansions centered at an origin are considered. It is known that the complete diagonal of such an expansion is an algebraic function. The order of a branch point of the diagonal by means of the logarithmic Gauss mapping of a polar curve of the rational function is described.

Flat modules over rings, acts over semigroups are modules or acts such that functor A⊗− preserves monomorphisms. A unar, i.e. a set with only an unary operation can be considered as an act over a free cycle semigroup. It is shown that a unar is flat if and only if it is a coproduct of unars, each of which is a line, ray or cycle.

The Grubbs's test statistics are studied, i.e. absolute values of extreme studentized deviations of n random observations from the mean. We consider the case when random observations have arbitrary continuous marginal distributions. The existence of two regions is proved; in one of them, the joint distribution function of these statistics is a linear function of their marginal distribution functions, and in the other, the joint distribution function is zero. We construct a Grubbs’s copula from the joint distribution of Grubbs’s statistics. For the case n>3, the existence of two domains within the unit square in which the Grubbs's copula coincides with the lower Frechet-Hoeffding boundary is proved. In the case of n=3, the Grubbs's copula is the Frechet-Hoeffding lower bound. The Grubbs's copula rotated by 180∘ also partially coincides with the Frechet-Hoeffding lower bound (in the case of n>3) and is the Frechet-Hoeffding lower bound (in the case of n=3). We prove that Grubbs's copulas rotated by 90∘ and 270∘ partially coincide with the Frechet-Hoeffding upper bound (in the case of n>3) and become the Frechet-Hoeffding upper bound (in the case n=3).