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Vol 84, No 4 (2020)
Articles
Viacheslav Valentinovich Nikulin (congratulation)
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(4):3-4
3-4
On orthogonal projections of Nöbeling spaces
Abstract
Suppose that $0\le k<\infty$. We prove that there is a dense open subset of the Grassmann space$\operatorname{Gr}(2k+1,m)$ such that the orthogonal projection of the standard Nöbeling space$N^m_k$ (which lies in $\mathbb R^m$ for sufficiently large $m$) to every $(2k+1)$-dimensional planein this subset is $k$-soft and possesses the strong $k$-universal property with respect to Polish spaces.Every such orthogonal projection is a natural counterpart of the standard Nöbeling space for the category of maps.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(4):5-40
5-40
Functions of perturbed pairs of non-commuting contractions
Abstract
We consider functions $f(T,R)$ of pairs of noncommuting contractions on Hilbert space and study the problem as to which functions $f$ we have Lipschitz type estimates in Schatten–von Neumann norms. We prove that if $f$ belongs to the Besov class $(B_{\infty,1}^1)_+(\mathbb{T}^2)$ of analytic functions in the bidisc, thenwe have a Lipschitz type estimate for functions $f(T,R)$ of pairs of not necessarily commuting contractions $(T,R)$ in the Schatten–von Neumann norms $\mathbf{S}_p$ for $p\in[1,2]$. On the other hand, we show that for functions in $(B_{\infty,1}^1)_+(\mathbb{T}^2)$, there are no such Lipschitz type estimates for $p>2$, nor in the operator norm.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(4):41-65
41-65
Displaying the cohomology of toric line bundles
Abstract
There is a standard approach to calculate the cohomology of torus-invariant sheaves$\mathcal{L}$ on a toric variety via the simplicial cohomology of the associated subsets$V(\mathcal{L})$ of the space $N_\mathbb{R}$ of 1-parameter subgroups of the torus.For a line bundle $\mathcal{L}$ represented by a formal difference $\Delta^+-\Delta^-$ of polyhedrain the character space $M_\mathbb{R}$, [1] contains a simpler formula for the cohomology of $\mathcal{L}$, replacing $V(\mathcal{L})$ by the set-theoretic difference $\Delta^- \setminus \Delta^+$.Here, we provide a short and direct proof of this formula.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(4):66-78
66-78
Sobolev spaces of functions on a Hilbert space endowed with a translation-invariant measure and approximations of semigroups
Abstract
We study measures on a real separable Hilbert space $E$ that are invariant undertranslations by arbitrary vectors in $E$. We define the Hilbert space $\mathcal H$ ofcomplex-valued functions on $E$ square-integrable with respect to some translation-invariant measure $\lambda$. We determine the expectations of the operators of shiftby random vectors whose distributions are given by semigroups (with respect toconvolution) of Gaussian measures on $E$. We prove that these expectations form a semigroup of self-adjoint contractions on $\mathcal H$. We obtain a criterion for thestrong continuity of such semigroups and study the properties of their generators(which are self-adjoint generalizations of Laplace operators to the case of functionsof infinite-dimensional arguments). We introduce analogues of Sobolev spaces andspaces of smooth functions and obtain conditions for the embedding and dense embedding ofspaces of smooth functions in Sobolev spaces. We apply these function spacesto problems of approximating semigroups by the expectations of random processes andstudy properties of our generalizations of Laplace operators and their fractionalpowers.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(4):79-109
79-109
Homogenization of Kirchhoff plates with oscillating edges and point supports
Abstract
We study deformations of a long (narrow after rescaling) Kirchhoff plate with periodic (rapidly oscillating) boundary.We deduce a limiting system of two ordinary differential equations of orders 4 and 2 which describe the deflection andtorsion of a two-dimensional plate in the leading order. We also consider point supports (Sobolev conditions) whoseconfiguration influences the result of homogenizing the biharmonic equation by decreasing the size of the limitingsystem of differential equations or completely eliminating it. The boundary-layer phenomenon near the endfaces of the plate is studied for various ways of fastening as well as for angular junctions of two long plates, possibly bypoint clamps (Sobolev conjugation conditions). We discuss full asymptotic series for solutions of static problems andthe spectral problems of plate oscillations.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(4):110-168
110-168
Proof of the Grothendieck–Serre conjecture on principal bundles over regular local rings containing a field
Abstract
Let $R$ be a regular local ring containing a field. Let $\mathbf{G}$ be a reductive group scheme over $R$.We prove that a principal $\mathbf{G}$-bundle over $R$ is trivial if it is trivial over the field of fractions of $R$.In other words, if $K$ is the field of fractions of $R$, then the map$$ H^1_{\mathrm{et}}(R,\mathbf{G})\to H^1_{\mathrm{et}}(K,\mathbf{G})$$of the non-Abelian cohomology pointed setsinduced by the inclusion of $R$ in $K$ has trivial kernel. This result was proved in [1] for regularlocal rings $R$ containing an infinite field.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(4):169-186
169-186
Integer expansion in systems of translates and dilates of a single function
Abstract
We study expansions with integer coefficients of elements in the multidimensional spaces$L_p\{(0,1]^m\}$, $1\leq p<\infty$, in systems of translates anddilates of a single function. We describe models useful in applications, including those in multimodular spaces.The proposed approximation of elements in $L_p\{(0,1]^m\}$, $1\leq p <\infty$, has the property of image compression, that is, there are many zero coefficients in this expansion. The studymay also be of interest to specialists in the transmission and processing ofdigital information since we find a simple algorithm for approximating in $L_p\{(0,1]^m\}$, $1 \leq p < \infty$, having this property.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(4):187-197
187-197
Existence and uniqueness of solution of a certain boundary-value problem for a convolution integral equation with monotone non-linearity
Abstract
We study the existence and uniqueness as well as the asymptotic behaviour ofsolutions of a certain boundary-value problem for a convolution integral equation on the whole line with monotone non-linearity. In some special cases, there are concrete applications to $p$-adic string theory, the mathematical theory of the geographical spread of an epidemic, the kinetic theory of gases and the theory of radiation transfer. We prove \linebreak the existence and uniqueness of an odd bounded continuous solution. The monotonicity and the integral asymptotics of this solution is also discussed. We finally give particular application-oriented examples of the equations considered, which illustrate the special nature of our results.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(4):198-207
198-207
General orthonormal systems and absolute convergence
Abstract
We study problems of the absolute convergence of Fourier series of differentiable functions with respect to general orthonormal systems (ONS). We obtain results which show that under certain conditions on functions in the ONS, the Fourier series of an arbitrary function in a certain differentiability class is absolutely convergent. We finally note that our results cannot be improved.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(4):208-220
208-220