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卷 84, 编号 3 (2020)
- 年: 2020
- 文章: 6
- URL: https://bakhtiniada.ru/1607-0046/issue/view/7544
Articles
On the rate of approximation in the unit disc of $H^1$-functions by logarithmic derivatives of polynomials with zeros on the boundary
摘要
We study uniform approximation in the open unit disc $D=ż\colon |z|<1\}$ by logarithmic derivatives of $C$-polynomials, that is, polynomials whose zeros lie on the unit circle $C=ż\colon |z| {=} 1\}$. We find bounds for the rate of approximation for functions in Hardy class $H^1(D)$ and certain subclasses. We prove bounds for the rate of uniform approximation (either in $D$ or its closure) by $h$-sums $\sum_k \lambda_k h(\lambda_k z)$ with parameters $\lambda_k\in C$.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(3):3-14
3-14
Blow-up instability in non-linear wave models with distributed parameters
摘要
We consider two model non-linear equations describing electric oscillations in systems with distributedparameters on the basis of diodes with non-linear characteristics. We obtain equivalent integral equations forclassical solutions of the Cauchy problem and the first and second initial-boundary value problemsfor the original equations in thehalf-space $x>0$. Using the contraction mapping principle, we prove the local-in-timesolubility of these problems. For one of these equations, we use the Pokhozhaev method of non-linear capacityto deduce a priori bounds giving rise to finite-time blow-up results and obtain upper bounds for the blow-uptime. For the other, we use a modification of Levine's method to obtain sufficient conditions for blow-upin the case of sufficiently large initial data and give a lower bound for the order of growth of a functionalwith the meaning of energy. We also obtain an upper bound for the blow-up time.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(3):15-70
15-70
Real Segre cubics, Igusa quartics and Kummer quartics
摘要
We prove some properties of real Segre cubics. In particular, we find the topological types of the real partsof Segre cubics as well as the topological types of the real parts of the complements of the Segre planes.We prove some differential-geometric properties of the real parts of real Segre cubics and Kummer quartics.We study the automorphism groups of real Segre cubics and, in particular, their action on the real parts ofthese cubics.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(3):71-118
71-118
On the arithmetic of modified idèle class groups
摘要
Let $k$ be a number field and $S$, $T$ sets of places of $k$. For each prime $p$, we define an invariant $\mathscr{G}=\mathscr{G}_p(k_\infty/k,S,T)$ related to the Galois group of the maximal abelian extension of $k$ which is unramified outside $S$ and splits completely in $T$. In the main theorem we interpret $\mathscr{G}$ in terms of another arithmetic object $\mathscr{U}$ that involves various unit groups and uses genus theory applied to certain modules,which are technically modified from idèle groups. We show that this interpretation is functorial with respect to $S$ and $T$ and thereby providesinteresting connections between $\mathscr{G}$ and $\mathscr{U}$ as $S$ and $T$ vary. The settings and methods are new, and different from the classical genus theoreticmethods for idèle groups. The advantage of the new methods at the finite level not only generalizes but also strengthens certain known results involving the maximal $p$-abelian profinite Galois groupof $k$ that is $S$-ramified and $T$-split in terms of the arithmetic of certain units of $k$. At the infinite level, the method relates the deep arithmeticof special units with those of profinite Galois groups. For example, for special cases of $S$ and $T$, the invariants $\mathscr{G}$ are related to the conjectures of Gross (or Kuz'min–Gross) and Leopoldtand accordingly, in these special cases, the functorial interpretation of $\mathscr{G}$ as $S$ and $T$ vary involves interestingconnections between the conjectures of Gross and Leopoldt in a simpler and more concrete way. As a result, we conjecture that $\mathscr{G}$ is finite for all finite disjoint sets$S$, $T$ over the cyclotomic $\mathbb{Z}_p$-tower of $k$, which includes the conjectures of Gross and Leopoldt as special cases.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(3):119-167
119-167
On a class of elliptic boundary-value problems with parameter and discontinuous non-linearity
摘要
We study an elliptic boundary-value problem in a bounded domain with inhomogeneous Dirichletcondition, discontinuous non-linearity and a positive parameter occurring as a factor in the non-linearity.The non-linearity is in the right-hand side of the equation. It is non-positive (resp. equal to zero) fornegative (resp, non-negative) values of the phase variable. Let $\widetilde{u}(x)$ be a solution ofthe boundary-value problem with zero right-hand side (the boundary function is assumed to be positive).Putting $v(x)=u(x)-\widetilde{u}(x)$, we reduce the original problem to a problem with homogeneousboundary condition. The spectrum of the transformed problem consists of the values of the parameterfor which this problem has a non-zero solution (the function $v(x)=0$ is a solution for all values of the parameter).Under certain additional restrictions we construct an iterative process converging to a minimal semiregularsolution of the transformed problem for an appropriately chosen starting point. We prove that any non-emptyspectrum of the boundary-value problem is a ray $[\lambda^*,+\infty)$, where $\lambda^*>0$. As an application,we consider the Gol'dshtik mathematical model for separated flows of an incompressible fluid. We show thatit satisfies the hypotheses of our theorem and has a non-empty spectrum.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(3):168-184
168-184
Asymptotics of approximation of continuous periodic functions by linear means of their Fourier series
摘要
We establish an asymptotic formula for the rate of approximation of Fourier series of individual periodic functions by linear averages with an error $\omega_{2m}(f;{1}/{n})$, $m\in\mathbb{N}$. This formula is applicable to the means of Riesz, Gauss–Weierstrass, Picard and others. The result is new even for the arithmetic means of partial Fourier sums. We use the formula to determine the asymptotic behaviour of functions in a certain class. Separately, we consider the case of positive integral convolution operators.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(3):185-202
185-202