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Vol 210, No 4 (2019)
Eigenvalue asymptotics of long Kirchhoff plates with clamped edges
Abstract
Asymptotic expansions are constructed for the eigenvalues and eigenfunctions of the Dirichlet problem for the biharmonic operator in thin domains (Kirchhoff plates with clamped edges). For a rectangular plate the leading terms are asymptotically determined from the Dirichlet problem for a second-order ordinary differential equation, while for a $\mathsf T$-junction of plates they are determined from another limiting problem in an infinite waveguide formed by three half-strips in the shape of a letter $\mathsf T$ and describing a boundary-layer phenomenon. Open questions are stated for which the method developed gives no answer. Bibliography: 33 titles.
Matematicheskii Sbornik. 2019;210(4):3-26
3-26
Groups of line and circle homeomorphisms. Criteria for almost nilpotency
Abstract
For finitely-generated groups of line and circle homeomorphisms a criterion for their being almost nilpotent is established in terms of free two-generator subsemigroups and the condition of maximality. Previously the author found a criterion for almost nilpotency stated in terms of free two-generator subsemigroups for finitely generated groups of line and circle homeomorphisms that are $C^{(1)}$-smooth and mutually transversal. In addition, for groups of diffeomorphisms, structure theorems were established and a number of characteristics of such groups were proved to be typical. It was also shown that, in the space of finitely generated groups of $C^{(1)}$-diffeomorphisms with a prescribed number of generators, the set of groups with mutually transversal elements contains a countable intersection of open dense subsets (is residual). Navas has also obtained a criterion for the almost nilpotency of groups of $C^{(1+\alpha)}$-diffeomorphisms of an interval, where $\alpha>0$, in terms of free subsemigroups on two generators. Bibliography: 21 titles.
Matematicheskii Sbornik. 2019;210(4):27-40
27-40
The foundations of $(2n,k)$-manifolds
Abstract
The focus of our paper is a system of axioms that serves as a basis for introducing structural data for $(2n,k)$-manifolds $M^{2n}$, where $M^{2n}$ is a smooth, compact $2n$-dimensional manifold with a smooth effective action of the $k$-dimensional torus $T^k$. In terms of these data a construction of a model space $\mathfrak E$ with an action of the torus $T^k$ is given such that there exists a $T^k$-equivariant homeomorphism $\mathfrak E\to M^{2n}$. This homeomorphism induces a homeomorphism $\mathfrak E/T^k\to M^{2n}/T^k$. The number $d=n-k$ is called the complexity of a $(2n,k)$-manifold. Our theory comprises toric geometry and toric topology, where $d=0$. It is shown that the class of homogeneous spaces $G/H$ of compact Lie groups, where $\operatorname{rk}G=\operatorname{rk}H$, contains $(2n,k)$-manifolds that have nonzero complexity. The results are demonstrated on the complex Grassmann manifolds $G_{k+1,q}$ with an effective action of the torus $T^k$. Bibliography: 23 titles.
Matematicheskii Sbornik. 2019;210(4):41-86
41-86
Convergence of spline interpolation processes and conditionality of systems of equations for spline construction
Abstract
This study is a continuation of research on the convergence of interpolation processes with classical polynomial splines of odd degree. It is proved that the problem of good conditionality of a system of equations for interpolation spline construction via coefficients of the expansion of the $k$th derivative in $B$-splines is equivalent to the problem of convergence of the interpolation process for the $k$th spline derivative in the class of functions with continuous $k$th derivatives. It is established that for interpolation with splines of degree $2n-1$, the conditions that the projectors corresponding to the derivatives of orders $k$ and $2n-1-k$ be bounded are equivalent.Bibliography: 26 titles.
Matematicheskii Sbornik. 2019;210(4):87-102
87-102
Linear collective collocation approximation for parametric and stochastic elliptic PDEs
Abstract
Consider the parametric elliptic problem $$-\operatorname{div}(a(y)(x)\nabla u(y)(x))=f(x),\qquad x\in D,\quad y\in\mathbb I^\infty,\quad u|_{\partial D}=0,$$where $D\subset\mathbb R^m$ is a bounded Lipschitz domain, $\mathbb I^\infty:=[-1,1]^\infty$, $f\in L_2(D)$, and the diffusion coefficients $a$ satisfy the uniform ellipticity assumption and are affinely dependent on $y$. The parameter $y$ can be interpreted as either a deterministic or a random variable. A central question to be studied is as follows. Assume that there is a sequence of approximations with a certain error convergence rate in the energy norm of the space $V:=H^1_0(D)$ for the nonparametric problem $-\operatorname{div}(a(y_0)(x)\nabla u(y_0)(x))=f(x)$ at every point $y_0\in\mathbb I^\infty$. Then under what assumptions does this sequence induce a sequence of approximations with the same error convergence rate for the parametric elliptic problem in the norm of the Bochner spaces $L_\infty(\mathbb I^\infty,V)$? We have solved this question using linear collective collocation methods, based on Lagrange polynomial interpolation on the parametric domain $\mathbb I^\infty$. Under very mild conditions, we show that these approximation methods give the same error convergence rate as for the nonparametric elliptic problem. In this sense the curse of dimensionality is broken by linear methods. Bibliography: 22 titles.
Matematicheskii Sbornik. 2019;210(4):103-127
103-127
128-144
Equivalence of the trigonometric system and its perturbations in the spaces $L^p$ and $C$
Abstract
Let $B=B[-\pi,\pi]$ be any of the spaces $L^p(-\pi,\pi)$, $1\leq p< \infty$, $p\neq2$, and $C[-\pi,\pi]$, and let $B_a=B[-\pi+a,\pi+a]$, $a\in\mathbb R$. A number of necessary conditions and sufficient conditions for the ‘perturbed trigonometric system’ $e^{i(n+\alpha_n)t}$, $n\in\mathbb Z$, to be equivalent to the trigonometric system $e^{int}$, $n\in\mathbb Z$, in the space $B_a$ for any $a\in\mathbb R$ are obtained. In particular, it is shown that if $(\alpha_n)\in l^s$, where $1/s=|1/p-1/2|$, then this equivalence takes place, the exponent $s$ being sharp. This result is used to show that in $L^p(-\pi,\pi)$, $1< p< 2$, there exist bases of exponentials which are not equivalent to the trigonometric basis.
The machinery of Fourier multipliers is used in the proofs.
Bibliography: 18 titles.
Matematicheskii Sbornik. 2019;210(4):145-164
145-164