In this paper, we construct a function U ∈L^1[0,1) that is uniformly greedy-universal with respect to the generalized Walsh system.
Several integral inequalities are presented. These inequalities generalize and refine the Gauss–Winckler and Faber inequalities for power moments from probability theory. The proofs are based on the method developed in our paper: F. G. Avkhadiev, “A simple proof of the Gauss–Winckler inequality,” Am. Math. Mon. 112, 459–462 (2005).
An algorithm with time complexity O( M(n)log n/loglog n) , for detecting a square number (with n digits) is developed from an algorithm that builds up the square root of a number (if it exists) from the low order digits to the high order digits. The algorithm depends on the properties of natural numbers in base 2^s , for any integer s ⩾ 3 . A large value of s (proportional to n ) is used to begin with, followed by progressively smaller values of s .
The article considers a boundary value problem of the Dirichlet type for a degenerate equation of high even order with singular coefficients. The solution to the problem is constructed as a series in eigenfunctions of the spectral problem for a degenerate ordinary differential equation of high even order with lower-order terms. In this case, a boundary value problem for a one-dimensional differential equation of high even order with lower-order terms is also solved by reducing it to a Fredholm integral equation of the second kind. Sufficient conditions for the solvability and uniqueness of the problem are obtained.
The paper is devoted to a homogenization of a Riemann–Hilbert problem for the Beltrami equation with local boundary conditions. Estimates of the homogenization error of the order O(√(ε)) ( q > 2 , ε > 0 is a small parameter), with a constant depending on the ellipticity constant and the region.
