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Vol 215, No 2 (2024)
Stable vector bundles and the Riemann–Hilbert problem on a Riemann surface
Abstract
The paper is devoted to holomorphic vector bundles with logarithmic connections on a compact Riemann surface and the applications of the results obtained to the question of solvability of the Riemann–Hilbert problem on a Riemann surface. We give an example of a representation of the fundamental group of a Riemann surface with four punctured points which cannot be realized as the monodromy representation of a logarithmic connection with four singular points on a semistable bundle. For an arbitrary pair of a bundle and a logarithmic connection on it we prove an estimate for the slopes of the associated Harder–Narasimhan filtration quotients. In addition, we present results on the realizability of a representation as a direct summand in the monodromy representation of a logarithmic connection on a semistable bundle of degree zero.
3-20
On an isometric embedding of prisms
Abstract
For an arbitrary convex polyhedral prism, a family of isometric embeddings of it is constructed that satisfy conditions similar to those that Pogorelov imposed on an isometry of a circular cylinder and called the ‘conditions of support on circles at the edges’.
21-32
The circle criterion and Tsypkin's criterion for systems with several nonlinearities without using the $S$-procedure
Abstract
The circle criterion (for continuous-time systems) and Tsypkin's criterion (for discrete-time systems) of absolute stability for Lurie systems with several nonlinearities are obtained with the use of the convolution theorem and without use of the S-procedure. On the basis of the convolution theorem, two theorems are proved which lead to a substantial reduction in the dimension of connected systems of linear matrix inequalities.
33-47
The exact univalent covering domain on the class of holomorphic self-maps of a disc with an interior and a boundary fixed points
Abstract
The class of holomorphic maps of the unit disc to itself, with an interior and a boundary fixed point is under consideration. For the class of such functions a sharp univalent covering domain is found in its dependence on the value of the angular derivative at the boundary fixed point and the position of the interior fixed point. This result can be viewed as a refinement of Landau's theorem on the univalent covering disc for the class of bounded holomorphic functions with prescribed derivative at the interior fixed point.
48-72
On quasi generic covers of the projective plane
Abstract
The Chisini Theorem for almost generic covers of the projective plane, whose proof is contained in the article “A Chisini Theorem for almost generic covers of the projective plane” (Sb. Math. 213:3 (2022), 341–356}), is generalized to the case of quasi-generic covers of the projective plane branched in curves with ADE-singularities.
73-102
The basis property of the Legendre polynomials in variable exponent Lebesgue space
Abstract
Sharapudinov proved that the Legendre polynomials form a basis of the Lebesgue space with variable exponent p(x) if p(x)>1 satisfies the Dini–Lipschitz condition and is constant near the endpoints of the orthogonality interval. We prove that the system of Legendre polynomials forms a basis of these spaces without the condition that the variable exponent be constant near the endpoints.
103-119
On capacities comparable to harmonic ones
Abstract
Let L be a second-order homogeneous elliptic differential operator in RN, N⩾3, with constant complex coefficients. Removable singularities of L∞-bounded solutions of the equation Lf=0 are described in terms of the capacities γL, where γΔ is the classical harmonic capacity from potential theory. It is shown for the corresponding values of N that γL and γΔ are commensurable for all L. Some ideas due to Tolsa are used in the proof. Various consequences of this commensurability are presented; in particular, criteria for the uniform approximation of functions by solutions of the equation Lf=0 are stated in terms of harmonic capacities.
120-146
Rate of convergence of Thresholding Greedy Algorithms
Abstract
The rate of convergence of the classical Thresholding Greedy Algorithm with respect to some bases is studied. We bound the error of approximation by the product of two norms, the norm of
147-162