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Vol 210, No 12 (2019)
- Year: 2019
- Articles: 7
- URL: https://ogarev-online.ru/0368-8666/issue/view/7460
The action of the Monge-Ampère operator on polynomials in the plane and its fixed points of polynomial type
Abstract
The action of the Monge-Ampère operator on polynomials of degree four in two variables is investigated. Two necessary conditions for the Monge-Ampère equation to have a solution are established. Sufficient conditions for solvability are indicated, which coincide with necessary conditions in certain cases. Invariant submanifolds of the action of the Monge-Ampère operator are found. Closed invariant chains of polynomials are constructed, and all the fixed points having the form of general polynomials of degree four are found.Bibliography: 9 titles.
Matematicheskii Sbornik. 2019;210(12):3-30
3-30
Symmetric semigroups with three generators
Abstract
In the theory of numerical semigroups the Frobenius problem of finding the largest integer that does not belong to the given semigroup plays an important role. The study of the Frobenius problem suggests distinguishing the class of symmetric semigroups, which have a quite simple structure. The main result in this work is an asymptotic formula describing the growth of the number of symmetric semigroups with three generators. Bibliography: 18 titles.
Matematicheskii Sbornik. 2019;210(12):31-42
31-42
Antisymmetric paramodular forms of weight 3
Abstract
The problem of the construction of antisymmetric paramodular forms of canonical weight 3 has been open since 1996. Any cusp form of this type determines a canonical differential form on any smooth compactification of the moduli space of Kummer surfaces associated to $(1,t)$-polarised abelian surfaces. In this paper, we construct the first infinite family of antisymmetric paramodular forms of weight $3$ as automorphic Borcherds products whose first Fourier-Jacobi coefficient is a theta block. Bibliography: 32 titles.
Matematicheskii Sbornik. 2019;210(12):43-66
43-66
The boundary values of solutions of an elliptic equation
Abstract
The paper is devoted to the study of the boundary behaviour of solutions of a second-order elliptic equation. Criteria are established for the existence of a boundary value of a solution of the homogeneous equation under the same conditions on the coefficients of the equation as were used to establish that the Dirichlet problem with a boundary function in $L_p$, $p>1$, has a unique solution. In particular, an analogue of Riesz's well-known theorem (on the boundary values of an analytic function) is proved: if a family of norms in the space $L_p$ of the traces of a solution on surfaces ‘parallel’ to the boundary is bounded, then this family of traces converges in $L_p$. This means that the solution of the equation under consideration is a solution of the Dirichlet problem with a certain boundary value in $L_p$. Estimates of the nontangential maximal function and of an analogue of the Luzin area integral hold for such a solution, which make it possible to claim that the boundary value is taken in a substantially stronger sense. Bibliography: 57 titles.
Matematicheskii Sbornik. 2019;210(12):67-97
67-97
Universality of $L$-Dirichlet functions and nontrivial zeros of the Riemann zeta-function
Abstract
We prove a joint discrete universality theorem for Dirichlet $L$-functions concerning joint approximation of a tuple of analytic functions by shifts $L(s+ih\gamma_k, \chi_1),…,L(s+ih\gamma_k,\chi_r)$, where $0<\gamma_1<\gamma_2<\dotsb$ is the sequence of imaginary parts of the nontrivial zeros of the Riemann zeta-function, $h$ is a fixed positive number, and $\chi_1,…,\chi_r$ are pairwise nonequivalent Dirichlet characters. We use a weak form of Montgomery's conjecture on the correlation of pairs of zeros of the Riemann zeta-function in the analysis. Moreover, we show the universality of certain compositions of Dirichlet $L$-functions with operators in the space of analytic functions. Bibliography: 31 titles.
Matematicheskii Sbornik. 2019;210(12):98-119
98-119
Analytic complexity of differential algebraic functions
Abstract
Examples of differential algebraic functions with infinite analytic complexity are constructed. The fact of their existence implies that the class of differential algebraic functions is wider than the class of functions with finite complexity. Bibliography: 6 titles.
Matematicheskii Sbornik. 2019;210(12):120-135
120-135
Some arithmetic properties of the values of entire functions of finite order and their first derivatives
Abstract
We describe a class of entire functions of finite order which, together with their first derivative, take sufficiently many algebraic values (with certain restrictions on the growth of the degree and height of these values). We show that, under certain conditions, any such function is a rational function of special form of an exponential. For entire functions of finite order which are not representable in the form of a finite linear combination of exponentials, we obtain an estimate for the number of points (in any fixed disc) at which the values of the function itself and its first derivative are algebraic numbers of bounded degree and height. Bibliography: 8 titles.
Matematicheskii Sbornik. 2019;210(12):136-150
136-150