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In this paper, we study the approximate control problem from the exterior of a nonlocal equation of Sobolev–Galpern type, specifically the Barenblatt–Zheltov–Kochina equation, involving the fractional Laplace operator of order $$s\in(0,1)$$ . We prove that the system under consideration is approximate controllable at any time $$T>0$$ .

At scattered places of his notebooks, Ramanujan recorded over 30 values of singular moduli $$\alpha_n$$ . All those results were proved by Berndt et. al by using Weber–Ramanujan’s class invariants. In this paper, we initiate to derive the explicit evaluations formula for $$\alpha_{9n}$$ and $$\alpha_{n/9}$$ by involving the class invariant. For this purpose, we establish several new $$P-Q$$ mixed modular equations involving theta-functions. We apply these modular equations further, deriving a new formula for the explicit evaluation of the Ramanujan–Selberg continued fraction.

In this paper, we introduce a new class of algebras satisfying certain conditions which has been named power algebras and we prove some fixed point theorems for a new class of contractions. We give new conditions for the existence and uniqueness of fixed points for this new class of mappings which are not a contraction in the usual algebra. Also, we introduce a new iterative algorithm for fixed-point problems in a Banach algebra. This results improve and extend the corresponding conclusions of the Ishikawa algorithm under weaker conditions and lead to stronger results. In conclusion, we give an application to nonlinear integral equations.

The paper is devoted to the study of conditional bounds for the expectation of the maximum of independent identically distributed standardized random variables for which the values of the skewness and kurtosis coefficients are known. With the aid of Holder’s inequality, an upper bound (in the form of a lower bound for a certain expression with parameters) is obtained and a criterion for the reachability of this estimate is formulated. A lower bound for the upper boundary of the expectation of the maximum is also found. A simpler and rougher upper bound is given in explicit form.

In the present paper, we propose a new method of derivation of number-theoretic identities which is applied to the proof of the multidimensional analogue of one of the Ramanujan identities. This method allows us to obtain new infinite series representations for the number $$\pi$$ , of the values of the Riemann zeta function, and of the $$L$$ -Dirichlet series at integer points.

Two examples illustrating properties of discrete measures are given. In the first part of the paper, it is proved that, for any probability measure $$\mu$$ with $$\operatorname{supp}{\mu}=[-1,1]$$ whose logarithmic potential is continuous on $$[-1,1]$$ , there exists a (discrete) measure $$\sigma=\sigma(\mu)$$ with $$\operatorname{supp}{\sigma}=[-1,1]$$ such that the corresponding orthogonal polynomials $$P_n(x;\sigma)=x^n+\dotsb$$ satisfy the condition $$(1/n)\chi(P_n(\,\cdot\,;\sigma))\xrightarrow{*}\mu$$ , $$n\to\infty$$ , where $$\chi(\,\cdot\,)$$ is the measure counting the zeros of a polynomial. The proof of the existence of such a measure $$\sigma$$ is based on properties of weighted Leja points. In the second part, an example of a compact set and a sequence of discrete measures supported on it with a special property is given. Namely, the sequence of measures converges in the $$*$$ -weak topology to the equilibrium measure on the compact set, but the corresponding sequence of logarithmic potentials converges in capacity to the equilibrium potential in no neighborhood of this compact set.

An algebra is called a binary Leibniz algebra if each of its two-generated subalgebras is a Leibniz algebra. In the present paper, we give a description of binary Leibniz algebras in terms of identities. As a consequence, we show that the variety of binary Leibniz algebras is not Schreier and that the freedom theorem fails to hold for this variety.

The paper deals with the problem of recovering solutions of a generalized Cauchy–Riemann system in a multidimensional spatial domain from their values on a piece of the boundary of this domain, i.e., an approximate solution of this problem based on the Carleman–Yarmukhamedov matrix method is constructed.

In Shallit’s problem (SIAM Review, 1994), it was proposed to justify a two-term asymptotics of the minimum of a rational function of $$n$$ variables defined as the sum of a special form whose number of terms is of order $$n^2$$ as $$n\to\infty$$ . Of particular interest is the second term of this asymptotics (“Shallit’s constant”). The solution published in SIAM Review presented an iteration algorithm for calculating this constant, which contained some auxiliary sequences with certain properties of monotonicity. However, a rigorous justification of the properties, necessary to assert the convergence of the iteration process, was replaced by a reference to numerical data. In the present paper, the gaps in the proof are filled on the basis of an analysis of the trajectories of a two-dimensional dynamical system with discrete time corresponding to the minimum points of $$n$$ -sums. In addition, a sharp exponential estimate of the remainder in Shallit’s asymptotic formula is obtained.

In this paper, the properties of the primitives of Legendre polynomials on the interval $$[0;1]$$ are studied. It is proved that the Legendre polynomials form an “almost” orthogonal system. Namely, for a fixed order of the primitive, only finitely many of these polynomials can be nonorthogonal. These properties underly the relationship between the spectral problems for differential operators in $$L_2[0;1]$$ and the spectral properties of generalized Jacobi matrices.