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Vol 85, No 3 (2021)
- Year: 2021
- Articles: 18
- URL: https://journals.rcsi.science/1607-0046/issue/view/7550
Articles
The issue is dedicated to the memory of Anatoliy Georgievich Vitushkin
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(3):3-4
3-4
Simple solutions of the Burgers and Hopf equations
Abstract
We describe all solutions of the Burgers equation of analytic complexity notexceeding $1$. It turns out that all such solutions fall into four families ofdimensions not exceeding $3$ that are represented by elementary functions. An example of a family of solutions of the Burgers equation of complexity $2$ is given.A similar problem is also solved for the Hopf equation. It turns out that allsolutions to the Hopf equation of complexity $1$ form a two-parameter family offractional-linear functions which coincides with one of the families of solutions ofthe Burgers equation.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(3):5-12
5-12
On a lower bound for the rate of convergence of multipoint Pade approximants of piecewise analytic functions
Abstract
We obtain a lower bound for the rate of convergence of multipoint Pade approximants of functions holomorphicallyextendable from a compact set to a union of domains whose boundaries possess a symmetry property. Thebound obtained matches a known upper bound for the same quantity.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(3):13-29
13-29
Tau functions of solutions of soliton equations
Abstract
In the holomorphic version of the inverse scattering method, we prove that the determinant of aToeplitz-type Fredholm operator arising in the solution of the inverse problem is an entire function of the spatial variablefor all potentials whose scattering data belong to a Gevrey class strictly less than 1. As a corollary, we establishthat, up to a constant factor,every local holomorphic solution of the Korteweg–de Vries equation is the second logarithmicderivative of an entire function of the spatial variable. We discuss the possible order of growth of this entire function.Analogous results are given for all soliton equations of parabolic type.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(3):30-51
30-51
52-72
Plane algebraic curves in fancy balls
Abstract
Boileau and Rudolph [1] called an oriented link $L$ in the 3-sphere a \textit{$\mathbb{C}$-boundary} if it can be realized as the intersection of an algebraic curve $A$ in $\mathbb{C}^2$ and the boundary of a smoothembedded closed 4-ball $B$. They showed that some links are not $\mathbb{C}$-boundaries. We say that $L$is a \textit{strong $\mathbb{C}$-boundary} if $A\setminus B$ is connected. In particular, all quasipositive links arestrong $\mathbb{C}$-boundaries.In this paper we give examples of non-quasipositive strong $\mathbb{C}$-boundaries and non-strong$\mathbb{C}$-boundaries. We give a complete classification of (strong) $\mathbb{C}$-boundaries with atmost five crossings.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(3):73-88
73-88
Uniform approximation of functionsby solutions of second order homogeneous strongly elliptic equations on compact sets in ${\mathbb{R}}^2$
Abstract
We obtain a criterion for the uniform approximability of functions by solutions of second-order homogeneous strongly ellipticequations with constant complex coefficients on compact sets in $\mathbb{R}^2$ (the particular case of harmonicapproximations is not distinguished).The criterion is stated in terms of the unique (scalar) Harvey–Polking capacity related to the leading coefficient of aLaurent-type expansion (this capacity is trivial in the well-studied case of non-strongly elliptic equations).The proof uses an improvement of Vitushkin's scheme, special geometric constructions, and methods of the theory of singular integrals. In view of the inhomogeneity of the fundamental solutions of strongly elliptic operatorson $\mathbb{R}^2$, the problem considered is technically more difficult than the analogous problemfor $\mathbb{R}^d$, $d>2$.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(3):89-126
89-126
On the Newton polyhedron of a Jacobian pair
Abstract
We introduce and describe the Newton polyhedron related to a “minimal” counterexample to the Jacobian conjecture. This description allows us to obtain a sharper estimate for the geometric degree of the polynomial mapping given by a Jacobian pair and to give a new proof in the case of the Abhyankar's two characteristic pairs.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(3):127-137
127-137
Explicit minimizers of some non-local anisotropic energies: a short proof
Abstract
In this paper we consider non-local energies defined on probability measures in the plane, given by a convolutioninteraction term plus a quadratic confinement. The interaction kernel is $-\log|z|+\alpha x^2/|z|^2$, $z=x+iy$, with $-1<\alpha<1$. This kernel is anisotropic except for the Coulomb case $\alpha=0$. We present a short compact proofof the known surprising fact that the unique minimizer of the energy is the normalized characteristic function of the domainenclosed by an ellipse with horizontal semi-axis $\sqrt{1-\alpha}$ and vertical semi-axis $\sqrt{1+\alpha}$.Letting $\alpha \to 1^-$, we find that the semicircle law on the vertical axis is the unique minimizer of the correspondingenergy, a result related to interacting dislocations, and previously obtained by some of the authors. We devote thefirst sections of this paper to presenting some well-known background material in the simplest way possible, so thatreaders unfamiliar with the subject find the proofs accessible.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(3):138-153
138-153
Criteria for $C^1$-approximability of functions on compact sets in ${\mathbb{R}}^N$, $N \geq 3$, by solutions of second-order homogeneous elliptic equations
Abstract
We obtain capacitive criteria for the approximability of individual functions by solutions of second-order homogeneous ellipticequations with constant complex coefficients in the norm of a Whitney-type $C^1$-space on a compact setin $\mathbb{R}^N$, $N \geq 3$. The case $N=2$ was studied in a recent paper by the author and Tolsa.For $C^1$-approximations by harmonic functions (with any $N$), weaker criteria were earlier found by the author.We establish some metric properties of the capacities considered.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(3):154-177
154-177
Quasi-polynomial mappings with constant Jacobian
Abstract
The famous Jacobian conjecture (JC) remains open even for dimension $2$. In this paper we study it by extending theclass of polynomial mappings to quasi-polynomial ones. We show that any possible non-invertible polynomialmapping with non-zero constant Jacobian can be transformed into a special reduced form by a sequence ofelementary transformations.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(3):178-190
178-190
On the classification of $3$-dimensional spherical Sasakian manifolds
Abstract
In this article we regard spherical hypersurfaces in $\mathbb{C}^2$ with a fixed Reeb vector field as $3$-dimensional Sasakian manifolds. We establish a correspondence between three different sets of parameters, namely, those arising from representing the Reeb vector field as an automorphism of the Heisenberg sphere, those used in Stanton's description of rigid spheres, and those arising from the rigid normal forms. We also describe geometrically the moduli space for rigid spheres and provide a geometric distinction between Stanton hypersurfaces and those found in [1]. Finally, we determine the Sasakian automorphism groups of rigid spheres and detect the homogeneous Sasakian manifolds among them.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(3):191-202
191-202
Holomorphically homogeneous CR-manifolds and their model surfaces
Abstract
We show that the model surface of a germ of a holomorphically homogeneousCR-manifold is holomorphically homogeneous. We also obtain restrictions on the multiplicitiesin the Bloom–Graham type of a germ of a holomorphically homogeneous CR-manifold.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(3):203-209
203-209
210-221
Approximation in measure: the Dirichlet problem, universality and the Riemann hypothesis
Abstract
We use approximation in measure to solve an asymptotic Dirichlet problem on arbitrary open sets and to show that many functions, including the Riemann zeta-function, are universal in measure. Connections with the Riemann hypothesis are suggested.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(3):222-238
222-238
Immersions of open Riemann surfaces into the Riemann sphere
Abstract
In this paper we show that the space of holomorphic immersions from any given open Riemann surface $M$into the Riemann sphere $\mathbb{CP}^1$ is weakly homotopy equivalent to the space of continuous maps from$M$ to the complement of the zero section in the tangent bundle of $\mathbb{CP}^1$. It follows in particular that thisspace has $2^k$ path components, where $k$ is the number of generators of the first homology group$H_1(M,\mathbb{Z})=\mathbb{Z}^k$. We also prove a parametric version of the Mergelyan approximation theoremfor maps from Riemann surfaces to an arbitrary complex manifold, a result used in the proof of our main theorem.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(3):239-260
239-260
Nevanlinna factorization in weighted classes of analytic functions of variable smoothness
Abstract
We define a new class of functions of variable smoothness that areanalytic in the unit disc and continuous in the closed disc. We construct the theoryof the Nevanlinna outer-inner factorization, taking into account the influence of theinner factor on the outer function, for functions of the new class.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(3):261-283
261-283
On compact subsets possessing strictly plurisubharmonic functions
Abstract
We give a geometric condition on a compact subset of a complex manifold which is necessary and sufficient for the existence of a smooth strictly plurisubharmonic function defined in a neighbourhood of this set.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(3):284-299
284-299