Two-dimensional autonomous systems of differential equations of the form ẋ= y P(x, y), ẏ = x + Q(x, y), where P and Q are holomorphic functions of order greater than or equal two are considered. In this work, necessary and sufficient conditions for the existence of an isochronous center or focus are obtained. These conditions are formulated in terms of commuting differential systems and some normal form.

In this paper, we investigate some boundary value problems for the Cauchy–Riemann equations in the triangular domain M. By the parqueting-reflection method, we obtain a covering of the complex plane through reflections of the domain. Then by the Cauchy–Pompeiu formula, we derive the Cauchy–Schwarz representation formula in M. In addition, we consider the Schwarz boundary value problem for the inhomogeneous Cauchy–Riemann equation and the explicit expressions of solution and condition of solvability. Finally, by using the Schwarz boundary value problem, the solution of the Dirichlet boundary value problem is explicitly solved.

We consider a Dirac-type operator of 2mth order on a finite interval G = (a, b). It is assumed that its coefficient is a complex-valued matrix function summable on G =, (a b). A Riesz property criterion is established for a system of root vector functions, and a theorem on the equivalent basis property in  L^2m_p (G), 1 < p < ∞  is prove

We consider the properties of systems $\Phi_1$ orthogonal with respect to a weighted discrete-continuous Sobolev inner product of the form $\langle f,g \rangle_S = f(a)g(a)+f(b)g(b)+\displaystyle\int_a^b f'(t)g'(t)w(t)dt$. The completeness of systems $\Phi_1$ in the Sobolev space $W^1_{L^2_w}$ and the relation of $\Phi_1$ to systems orthogonal in weighted Lebesgue spaces $L^2_u$ are studied. We also analyze properties of the Fourier series with respect to systems $\Phi_1$. In particular, conditions for the uniform convergence of Fourier series to functions from $W^1_{L^2}$ are obtained.

Certain new estimates for the coefficients of a polynomial are obtained, which among other things furnishes new but equivalent forms of some well-known results of V.K. Jain and S. Gulzar.

We consider Haar-type systems, which are generated by a (generally speaking, unbounded) sequence $ \{ p_n \}_{n=1}^\infty $, and which are defined on the modified segment $ [0, 1]^* $, i.\,e., on the segment [0, 1] whose $ \{ p_n \}$-rational points are calculated two times. The main result of this work is a Jordan-type test for the pointwise and uniform convergence of Fourier series with respect to Haar-type systems. It is shown that the test obtained in the paper can not be improved. An example of a function of bounded variation, whose Fourier series with respect to Haar-type system diverges at some point, is constructed in the case of $ \sup\limits_n p_n = \infty$. Thus for any unbounded sequence $ \{ p_n \}_{n=1}^\infty $ there exists a monotone function whose Fourier series with respect to Haar-type system, generated by the given sequence $ \{ p_n \}_{n=1}^\infty $, diverges at some point. It is found that the Jordan test of convergence of Fourier series with respect to Haar-type systems does not differ from the Dini--Lipschitz condition for those systems. As the Dini--Lipschitz condition was considered earlier, the main value of this work is the construction of a corresponding counterexample, i.\,e., the construction of an example (a model) of a function of bounded variation whose Fourier series with respect to Haar-type Systems diverges at some point. In the counterexamples from the earlier works on the Dini--Lipschitz criterion (as well as for all Dini convergence criteria), the functions were {\it not of bounded variation}. The article's conclusion discusses how the Jordan convergence criterion evolved when transitioning from trigonometric systems of functions to Price systems (and to N.Ya. Vilenkin systems), and from there to generalized Haar systems and Haar-type systems.

We describe the shortest polygonal chains that connect two points on the first Heisenberg group with the sub-Riemannian structure. The shortest polygonal chain connecting two points with a fixed number of links either is a straight line or consists of segments of the same length such that the projections of their endpoints are inscribed in a circle. The analytical description is obtained for the spheres of the quasimetric generated by the shortest polygonal chains with 3 links.

The equivalence of two variants of problem formulation for rod-strip mechanics connected to a rigid and fixed support element on a finite-size section of one of its face surfaces is proved. The first of them is based on the use of a transformation model of rod deformation based on the transformation of the relations of the simplest Timoshenko shear model by preliminary satisfying the conditions of kinematic conjugation of the rod with the support element on the fastening section with the subsequent formulation of the kinematic and force conditions of conjugation of the fixed section of the rod with the unfastened one. The second variant, corresponding to the contact formulation of the problem, is based on the use of uniform relations of the refined theory of the Timoshenko type for the entire rod (for the fixed and unfastened sections), containing an unknown tangential contact stress at the points of the surface of the connection of the rod with the support element on the fixed section, which is included in the equations as an external load. To determine this, the conditions of kinematic conjugation of the rod with the support element are used.

The paper proposes an approach that makes it possible to describe in the same way classical Sobolev spaces, various generalizations of classical Sobolev spaces for functions defined on a metric or topological space and for functions with values in a metric or topological space, as well as spaces of functions that satisfy certain differential relations.

. We provide a criterion for the equivalency of the second-kind involutions of upper triangular matrix algebras over commutative rings. For an algebra T_n(F ) of upper triangular matrices over a field F it is proven that two involutions are equivalent if and only if they coincide after restriction to F .