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Vol 212, No 11 (2021)
- Year: 2021
- Articles: 7
- URL: https://journals.rcsi.science/0368-8666/issue/view/7495
New moduli components of rank 2 bundles on projective space
Abstract
We present a new family of monads whose cohomology is a stable rank 2 vector bundle on $\mathbb{P}^3$. We also study the irreducibility and smoothness together with a geometrical description of some of these families. These facts are used to construct a new infinite series of rational moduli components of stable rank 2 vector bundles with trivial determinant and growing second Chern class. We also prove that the moduli space of stable rank 2 vector bundles with trivial determinant and second Chern class equal to 5 has exactly three irreducible rational components.Bibliography: 40 titles.
Matematicheskii Sbornik. 2021;212(11):3-54
3-54
Orthogonality in nonseparable rearrangement-invariant spaces
Abstract
Let $E$ be a nonseparable rearrangement-invariant space and let $E_0$ be the closure of the space of bounded functions in $E$. Elements of $E$ orthogonal to $E_0$, that is, elements $x\in E$, $x\ne 0$, such that $\|x\|_{E} \le\|x+y\|_{E}$ for each $y\in E_0$, are investigated. The set of orthogonal elements $\mathcal{O}(E)$ is characterized in the case when $E$ is a Marcinkiewicz or an Orlicz space. If an Orlicz space $L_M$ is considered with the Luxemburg norm, then the set $L_M\setminus (L_M)_0$ is the algebraic sum of $\mathcal{O}(L_M)$ and $(L_M)_0$. Each nonseparable rearrangement-invariant space $E$ such that $\mathcal{O}(E)\ne\varnothing$ is shown to contain an asymptotically isometric copy of the space $l_\infty$. Bibliography: 17 titles.
Matematicheskii Sbornik. 2021;212(11):55-72
55-72
A probability estimate for the discrepancy of Korobov lattice points
Abstract
Bykovskii (2002) obtained the best current upper estimate for the minimum discrepancy of the Korobov lattice points from the uniform distribution. We show that this estimate holds for almost all $s$-dimensional Korobov lattices of $N$ nodes, where $s\ge 3$, and $N$ is a prime number. Bibliography: 14 titles.
Matematicheskii Sbornik. 2021;212(11):73-88
73-88
On optimal recovery of values of linear operators from information known with a stochastic error
Abstract
The optimal recovery of values of linear operators is considered for classes of elements the information on which is known with a stochastic error. Linear optimal recovery methods are constructed that, in general, do not use all the available information for the measurements. As a consequence, an optimal method is described for recovering a function from a finite set of its Fourier coefficients specified with a stochastic error. Bibliography: 14 titles.
Matematicheskii Sbornik. 2021;212(11):89-108
89-108
Estimates for the volume of the zeros of a holomorphic function depending on a complex parameter
Abstract
Given a holomorphic function $f(\sigma,z)$, $\sigma\in\mathbb{C}^{m}$, $z\in\mathbb{C}^{n}$, an estimate for the volume of the zero set $ż\colon f(\sigma,z)=0\}$ is presented which holds uniformly in $\sigma $. Such estimates are quite useful in investigations of oscillatory integrals of the form $$ J(\lambda,\sigma)=\int_{\mathbb{R}^{n} }a(\sigma, x)e^{i\lambda \Phi (\sigma, x)} dx $$ as $\lambda \to \infty $. Here $a(\sigma, x)\in C_{0}^{\infty } (\mathbb{R}^{n} \times\mathbb{R}^{m})$ is a so-called amplitude function and $\Phi (\sigma, x)$ is a phase function. Bibliography: 9 titles.
Matematicheskii Sbornik. 2021;212(11):109-115
109-115
Global boundedness of functions of finite order that are bounded outside small sets
Abstract
We prove that subharmonic or holomorphic functions of finite order on the plane, in space, or on the unit disc or ball that are bounded above on a sequence of circles or spheres, or on a system of embedded discs or balls, outside some asymptotically small sets are bounded above throughout. Hence, subharmonic functions of finite order on the complex plane, entire or plurisubharmonic functions of finite order, and also convex or harmonic functions of finite order that are bounded above on spheres outside such sets are constants. The results and the approaches to the proofs are new for both functions of one and several variables. Bibliography: 14 titles.
Matematicheskii Sbornik. 2021;212(11):116-127
116-127
Convergence of two-point Pade approximants to piecewise holomorphic functions
Abstract
Let $f_0$ and $f_\infty$ be formal power series at the origin and infinity, and $P_n/Q_n$, $\deg(P_n),\deg(Q_n)\leq n$, be the rational function that simultaneously interpolates $f_0$ at the origin with order $n$ and $f_\infty$ at infinity with order ${n+1}$. When germs $f_0$ and $f_\infty$ represent multi-valued functions with finitely many branch points, it was shown by Buslaev that there exists a unique compact set $F$ in the complement of which the approximants converge in capacity to the approximated functions. The set $F$ may or may not separate the plane. We study uniform convergence of the approximants for the geometrically simplest sets $F$ that do separate the plane. Bibliography: 26 titles.
Matematicheskii Sbornik. 2021;212(11):128-164
128-164