Let $G$ be a group and the set of primes $\tau(G)=\cup\pi(G : M)$ for any maximal subgroup $M$ of $G$. For a non-empty nilpotent formation $\mathfrak{X}$, it is proved that a group $G$ has a nilpotent $\mathfrak{X}$-residual if and only if the $\mathfrak{X}$-residual of the $p$-Sylow normalizer is subnormal in $G$ for every $p$ from $\tau(G)$.

We consider properties of polynomials with coefficients in division rings. A theorem on the decomposition of a polynomial with coefficients in an arbitrary division ring is obtained. It is shown that if a non-central element is not a root of a polynomial over an arbitrary division ring, then the conjugacy class of this element contains infinitely many elements that are not roots of this polynomial. The paper also contains estimates for the number of different conjugacy classes of spherical roots for some types of polynomials over quaternion algebras.

Modular secret sharing in the group $SL_2(\mathbb{Z})$ was recently proposed by Yanchevskiy, Matveev, and Govorushko. In this paper we have constructed in explicit form the entire fundamental domain under the action of left shifts of the principal congruence subgroup on the group $SL_2(\mathbb{Z})$, which presents additional possibilities for constructing schemes, since the domain is the space of stored secrets of the secret sharing scheme.

For finite $\pi$-solvable non-$\pi$-closed irreducible complex linear groups of degree $n$ with an $\pi$-Hall $TI$-subgroup $H$ of odd order greater than 3 conditions are found under which $n$ is divisible by $|H|$ or by a power $f > 1$ of some prime number such that $f\equiv 1(\mathrm{mod}\ |H|)$.

Let $\mathfrak{F}$ and $\mathfrak{H}$ be some $\tau$-closed $\sigma$-local formations of finite groups. By $\mathfrak{F}/^\tau_\sigma\mathfrak{H}\cap\mathfrak{F}$ we denote the lattice of all $\tau$-closed $\sigma$-local formations $\mathfrak{X}$ such that $\mathfrak{H}\cap\mathfrak{F}\subseteq \mathfrak{X}\subseteq \mathfrak{F}.$ The length of the lattice $\mathfrak{F}/^\tau_\sigma\mathfrak{H}\cap\mathfrak{F}$ is called the {\it $\mathfrak H^\tau_\sigma$-defect}, and for $\mathfrak{H}=(1)$ it is the formation of all identity groups, {\it $l^\tau_\sigma$-length} of $\mathfrak{F}$. The general properties of the $\mathfrak H^\tau_\sigma$-defect of $\tau$-closed $\sigma$-local formations are studied, and a description of the structural structure of reducible $\tau$-closed $\sigma$-local formations with $\mathfrak H^\tau_\sigma$-defect $\leq 2$ and $l^\tau_\sigma$-length $\leq 3$ is obtained.

Let $X=\{X_k\}_{k=1}^\infty$ be a sequence of independent symmetric bounded random variables. This paper investigates systems of the form $\{X_iX_j\}_{i<j}$, $\{X_i X_j X_k\}_{i<j<k},\ldots$, finite unions of such systems, and systems close to them, in the space $L_\infty$ of bounded random variables. Series over such systems do not hold the property of unconditionality: the convergence of the series depends on the ordering of the terms. At the same time, as we demonstrate in the paper, such systems posess a very close property of random unconditional convergence (or RUC-property).

The rational Jackson singular integral is introduced, which is a linear combination of Fourier-Chebyshev rational integral operators with a corresponding triangular matrix of coefficients and a fixed number of geometrically different poles. Its integral representation is established. Rational approximations of Markov functions on the segment $[-1,1]$ are investigated by the introduced method. An integral representation of approximations and an upper bound of uniform approximations are established. Approximations of Markov functions with an absolutely continuous measure whose derivative is asymptotically equal to a function with a power singularity are studied. In this case, top-down estimates of pointwise and uniform approximations and an asymptotic expression of the majorant of uniform approximations are found. Optimal values of the parameters of rational Jackson singular integrals are established, at which the best uniform approximations of Markov functions are provided by this method. For this purpose, the corresponding extreme problem is solved. It is shown that with a special choice of parameters, uniform rational approximations have a higher rate of decrease in comparison with the corresponding polynomial analogues. As a corollary, approximations of some elementary functions represented by Markov functions on the segment $[-1,1]$ are considered.

A linear periodic control system with a constant control matrix is considered. The program control is periodic, and its period is incommensurate with the period of the coefficient matrix. The feasible set of such periodic controls is called irregular. The problem is posed of selecting a control from this feasible set so that the now quasiperiodic system has a periodic solution with a given frequency spectrum whose period coincides with the control period. This problem is called the asynchronous spectrum control problem with an irregular feasible set. A necessary condition for its solvability is given.

A two-dimensional anti-Perron effect of changing arbitrary different positive Lyapunov exponents of a linear differential system to negative ones by a perturbation of a higher order of smallness is realized.

A new linear integro-differential equation is studied on a closed curve located on the complex plane. There are some restrictions on the curve and the coefficients of the equation. The equation contains hypersingular integrals with the desired function. A characteristic feature of the equation is the presence of regular integrals with the desired function and its complex conjugate value. The solution of the equation is reduced to solving a mixed boundary value problem for analytic functions and the subsequent solutijn of differential equations with additional conditions on the solution. The conditions for the solvability of the original equation are explicitly stated. When these are performed, the solution is in closed form. An example is given.

We consider a queueing network with negative customers, single-server nodes, and constraints on the sojourn time of customers in nodes. If, at the moment a negative customer arrives at a node, there are positive customers present, one of the positive customers instantly disappears from the network. If, however, no positive customers are present in the node at that moment, the incoming negative customer vanishes immediately and has no further effect on the network’s behavior. Positive customers whose sojourn time in a node has expired instantly and independently of other positive customers begin routing according to a transition matrix that differs from the routing matrix used by positively served customers. The insensitivity of the stationary distribution to the shape of the service time distribution given fixed first moments is proven. The conditional distribution of customer sojourn times in nodes is exponential.