Takagi function is a simple example of a continuous but nowhere differentiable function. It is defined by T(x) = ∞ ᢘ k=0 2−nρ(2nx), 
where ρ(x) = min k∈Z |x − k|. The moments of Takagi function are defined as Mn = ᝈ 1 0  xnT(x) dx. The main result of this paper is the following: Mn = lnn − Γ᝘(1) − lnπ n2 ln 2  + 1 2n2 + 2 n2 ln 2 φ(n) + O(n−2.99), where φ(x) = ᝨ kᡘ=0 Γ ᝈ2πik  ln 2 ᣸ ζ ᡸ2πik ln 2  ᡸ x−2lπni2k .

The movement of an object characterized by ordinary differential equations (ODE) with discontinuous right-hand sides along the surface of a gap is called a sliding mode. It is required to find the connection of the right-slip characteristics of the system (the system to continue the solution on the surface of the gap). The article prompted a sequel based on the solution of the averaged optimization. It is shown that for the known examples of methods for solving optimization averaged lead to results coinciding with the method of A.F. Filippov and allow to extend these techniques to a wide class of multidimensional problems. Optimality conditions set forth averaged nonlinear programming and examples of their use in the case of ordinary and degenerate solutions.

We consider several families of combinatorial polytopes associated with the following NP-complete problems: maximum cut, Boolean quadratic programming, quadratic linear ordering, quadratic assignment, set partition, set packing, stable set, 3-assignment. For comparing two families of polytopes we use the following method. We say that a family

For singularly perturbed second order equations the dependence of eigenvalues of the first boundary problem on a small parameter at the highest derivative is studied. The main assumption is that the coefficient at the first derivative in the equation is the sign of the variable. This leads to the emerging of so-called turning points. Asymptotic expansions on the small parameter are obtained for all eigenvalues of the considered boundary problem. It turns out that the expansions are defined by the behavior of coefficients in a neighborhood of turning points only

For a second order equation with a small factor at the highest derivative the asymptotic behavior of all eigenvalues of periodic and antiperiodic problems is studied. The main assumption is that the coefficient at the first derivative in the equation is the sign of the variable so that turning points exist an algorithm for computing all coefficients of asymptotic series for every considered eigenvalue is developed. It turns out that the values of these coefficients are defined by coefficient values of the original equation only in a neighborhood of turning points. Asymptotics for the length of Lyapunov zones of stability and instability was obtained. In particular, the problem of stability of solutions of second order equations with periodic coefficients and small parameter at the highest derivative was solved

We correct the computational mistakes in paper: Nesterov P. N. Center Manifold Method in the Asymptotic Integration Problem for Functional Differential Equations with Oscillatory Decreasing Coefficients. II. In: Modeling and Analysis of Information Systems. 2014. Vol. 21, № 5. P. 5 – 37, (in Russian).