Ашық рұқсат
Рұқсат берілді
Тек жазылушылар үшін
Том 85, № 2 (2021)
- Жылы: 2021
- Мақалалар: 7
- URL: https://ogarev-online.ru/1607-0046/issue/view/7549
Articles
On a class of Anosov diffeomorphisms on the infinite-dimensional torus
Аннотация
We study a quite natural class of diffeomorphisms $G$ on $\mathbb{T}^{\infty}$, where $\mathbb{T}^{\infty}$ is the infinite-dimensional torus (the direct product of countably many circles endowed with the topology of uniform coordinatewise convergence). The diffeomorphisms under consideration can be represented as the sums of a linear hyperbolic map and a periodic additional term. We find some constructive sufficient conditions, which imply that any $G$ in our class is hyperbolic, that is, an Anosov diffeomorphism on $\mathbb{T}^{\infty}$. Moreover, under these conditions we prove the following properties standard in the hyperbolic theory: the existence of stable and unstable invariant foliations, the topological conjugacy to a linear hyperbolic automorphism of the torus and the structural stability of $G$.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(2):3-59
3-59
General Fourier coefficients and convergence almost everywhere
Аннотация
We find sufficient conditions which are in a sense best possible that must be satisfied by the functions of an orthonormal system $(\varphi_n)$ in order for the Fourier coefficients of functions of bounded variation to satisfy the hypotheses of the Men'shov–Rademacher theorem. We also prove a theorem saying that every system $(\varphi_n)$ contains a subsystem $(\varphi_{n_k})$ with respect to which the Fourier coefficients of functions of bounded variation satisfy those hypotheses. Theresults obtained complement and generalize the corresponding results in [1].
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(2):60-72
60-72
Functions universal with respect to the trigonometric system
Аннотация
We construct an integrable function whose Fourier series possesses the following property. After an appropriatechoice of signs of the coefficients of this series, the partial sums of the resulting series are dense in $L^p$, $p\in(0,1)$.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(2):73-94
73-94
Positive solutions of superlinear elliptic problems with discontinuous non-linearities
Аннотация
We consider an elliptic boundary-value problem with a homogeneous Dirichlet boundary condition,a parameter and a discontinuous non-linearity. The positive parameterappears as a multiplicative term in the non-linearity, and the problemhas a zero solution for any value of the parameter. The non-linearity hassuperlinear growth at infinity. We prove the existence of positive solutions by a topological method.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(2):95-112
95-112
On a real caustic of type $E_6$
Аннотация
We prove that the manifold of non-singular points of a stable real caustic germ of type $E_6$and the manifolds of points of transversal intersection of its smooth branches consist only of contractibleconnected components. We also calculate the number of these components.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(2):113-141
113-141
Properties of monotone path-connected sets
Аннотация
We study monotone path-connected sets and also strongly and weakly Menger-connected sets. We introducethe notion of $\varepsilon$-solarity and establish a connection with the notion of solarity. We prove that boundedlycompact suns in $C(Q)$ are monotone path-connected.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(2):142-171
142-171
Exact value of the exponent of convergence of the singular integral in Tarry's problem for homogeneous polynomials of degree $n$ in two variables
Аннотация
Jabbarov [1] obtained the exact value of the exponent of convergence of the singular integral in Tarry's problemfor homogeneous polynomials of degree $2$. We extend this result to the case of polynomials of degree $n$.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(2):172-180
172-180