Generalization of the Hardy-Littlewood theorem on Fourier series

In the theory of one-dimensional trigonometric series, the Hardy-Littlewood theorem on Fourier series with monotone Fourier coefficients is of great importance. Multidimensional versions of this theorem have been extensively studied for the Lebesgue space. Significant differences of the multidimensional variants in comparison with the one-dimensional case are revealed and the strengthening of this theorem is obtained. The Hardy-Littlewood theorem is also generalized for various function spaces and various types of monotonicity of the series coefficients. Some of these generalizations can be seen in works of M.F. Timan, M.I. Dyachenko, E.D. Nursultanov, S. Tikhonov. In this paper, a generalization of the Hardy-Littlewood theorem for double Fourier series of a function in the space L_qϕ(L_q)(0,2π]^2 is obtained.

Homogeneous Euler equation: blow-ups, gradient catastrophes and singularity of mappings

The paper is devoted to the analysis of the blow-ups of derivatives, gradient catastrophes and dynamics of mappings of ℝn → ℝn associated with the n-dimensional homogeneous Euler equation. Several characteristic features of the multi-dimensional case (n > 1) are described. Existence or nonexistence of blow-ups in different dimensions, boundness of certain linear combinations of blow-up derivatives and the first occurrence of the gradient catastrophe are among of them. It is shown that the potential solutions of the Euler equations exhibit blow-up derivatives in any dimenson n. Several concrete examples in two- and three-dimensional cases are analysed. Properties of ℝnu → ℝ nx mappings defined by the hodograph equations are studied, including appearance and disappearance of their singularities.

Batalin–Vilkovisky quantization of fuzzy field theories

We apply the modern Batalin–Vilkovisky quantization techniques of Costello and Gwilliam to noncommutative field theories in the finite-dimensional case of fuzzy spaces. We further develop a generalization of this framework to theories that are equivariant under a triangular Hopf algebra symmetry, which in particular leads to quantizations of finite-dimensional analogues of the field theories proposed recently through the notion of ‘braided $$L_\infty $$ L ∞ -algebras’. The techniques are illustrated by computing perturbative correlation functions for scalar and Chern–Simons theories on the fuzzy 2-sphere, as well as for braided scalar field theories on the fuzzy 2-torus.

Mathematical modeling of water purification in a bioplato filter

A mathematical model of filtration taking into account clogging and suffusion in the bioplato filter system in the two-dimensional case was built. The constructed mathematical model takes into account the physical effects of the dynamic change of porosity and the dependence of the filtration coefficient on the concentration of contaminants, which is not in the known analogues

On -genericity of distributional chaos

Let M be a compact smooth manifold without boundary. Based on results by Good and Meddaugh [Invent. Math.220 (2020), 715–736], we prove that a strong distributional chaos is $C^0$ -generic in the space of continuous self-maps (respectively, homeomorphisms) of M. The results contain answers to questions by Li, Li and Tu [Chaos26 (2016), 093103] and Moothathu [Topology Appl.158 (2011), 2232–2239] in the zero-dimensional case. A related counter-example on the chain components under shadowing is also given.

Examples of S—expansions of Lie Algebras

In this note, we give examples of S—expansions of Lie algebras of finite and infinite dimension. For the finite dimensional case, we expand all real three-dimensional Lie algebras. In the case of infinite dimension, we perform contractions obtaining new Lie algebras of infinite dimension.

Solving the hourglass instability problem using rare mesh variation-difference schemes

The hourglass instability effect is characteristic of the Wilkins explicit difference scheme or similar schemes based on two-dimensional 4-node or three-dimensional 8-node finite elements with one integration point in the element. The hourglass effect is absent in schemes with cells in the form of simplexes (triangles in two-dimensional case, tetrahedrons in three-dimensional case). But they have another well-known drawback - slow convergence. One of the authors proposed a rare mesh scheme, in which elements in the form of a tetrahedron are located one at a time in the centers of the cells of a hexahedral grid. This scheme showed the absence “hourglass” effect and other drawbacks with high efficiency. This approach was further developed for solving 2D and 3D problems.

Approximate methods for solving amplitude-phase problem for discrete signals

Methods for solving amplitude and phase problems for one and two-dimensional discrete signals are proposed. Methods are based on using nonlinear singular integral equations. In the one-dimensional case amplitude and phase problems are modeled by corresponding linear and nonlinear singular integral equations. In the two-dimensional case amplitude and phase problems are modeled by corresponding linear and nonlinear bisingular integral equations. Several approaches are presented for modeling two-dimensional problems: 1) reduction of amplitude and phase problems to systems of linear and nonlinear singular integral equations; 2) using methods of the theory of functions of many complex variables, problems are reduced to linear and nonlinear bisingular integral equations. To solve the constructed nonlinear singular integral equations, methods of collocation and mechanical quadrature are used. These methods lead to systems of nonlinear algebraic equations, which are solved by the continuous method for solution of nonlinear operator equations. The choice of this method is due to the fact that it is stable against perturbations of coefficients in the right-hand side of the system of equations. In addition, the method is realizable even in cases where the Frechet and Gateaux derivatives degenerate at a finite number of steps in the iterative process. Some model examples have shown effectiveness of proposed methods and numerical algorithms.