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Vol 86, No 6 (2022)
- Year: 2022
- Articles: 11
- URL: https://journal-vniispk.ru/1607-0046/issue/view/7559
Articles
Congratulations to Evgenii Mikhailovich Chirka
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2022;86(6):3-4
3-4
Functions of class $C^\infty$ in non-commuting variablesin the context of triangular Lie algebras
Abstract
We construct a certain completion $C^\infty_\mathfrak{g}$ of the universalenveloping algebra of a triangular real Lie algebra $\mathfrak{g}$.It is a Frechet–Arens–Michael algebra that consists of elementsof polynomial growth and satisfies to the following universal property:every Lie algebra homomorphism from $\mathfrak{g}$ to a real Banach algebraall of whose elements are of polynomial growth has an extensionto a continuous homomorphism with domain $C^\infty_\mathfrak{g}$.Elements of this algebracan be calledfunctions of class $C^\infty$ in non-commuting variables.The proof is based on representation theory and employsan ordered $C^\infty$-functional calculus. Beyond the general case,we analyze two simple examples. As an auxiliary material, the basicsof the general theory of algebras of polynomial growthare developed. We also consider local variants of the completion and obtaina sheaf of non-commutative functions on the Gelfand spectrumof $C^\infty_\mathfrak{g}$ in the case when $\mathfrak{g}$ is nilpotent.In addition, we discuss the theory of holomorphic functions in non-commutingvariables introduced by Dosi andapply our methods to prove theorems strengthening some his results.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2022;86(6):5-46
5-46
Strong convergence of attractors of reaction-diffusion system with rapidly oscillatingterms in an orthotropic porous medium
Abstract
A system of reaction-diffusion equations in a perforated domain with rapidlyoscillating terms in the equations and in the boundary conditions isconsidered. It is not assumed that the uniqueness theorem conditions are satisfied for the corresponding initial-boundary value problem. We have proved the strong convergence of the trajectory attractors of this system to thetrajectory attractors of the homogenized reaction-diffusion system with a ‘strange term’ (potential).
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2022;86(6):47-78
47-78
When is the search of relatively maximal subgroups reduced to quotient groups?
Abstract
Let $\mathfrak{X}$ be a class finite groups closed under taking subgroups, homomorphic images, and extensions, andlet $\mathrm{k}_{\mathfrak{X}}(G)$ be the number of conjugacy classes $\mathfrak{X}$-maximal subgroups of a finite group $G$.The natural problem calling for a description, up to conjugacy, ofthe $\mathfrak{X}$-maximal subgroups of a given finite group is not inductive.In particular, generally speaking, the image of an $\mathfrak{X}$-maximalsubgroup is not $\mathfrak{X}$-maximal in the image of a homomorphism.Nevertheless, there exist group homomorphisms that preserve the number of conjugacy classes of maximal$\mathfrak{X}$-subgroups (for example, the homomorphisms whose kernels are $\mathfrak{X}$-groups).Under such homomorphisms, the image of an $\mathfrak{X}$-maximal subgroup is always $\mathfrak{X}$-maximal,and, moreover, there is a natural bijection between the conjugacy classesof $\mathfrak{X}$-maximal subgroups of the image and preimage.In the present paper, all such homomorphisms arecompletely described.More precisely, it is shown that, for a homomorphism $\phi$from a group $G$, the equality $\mathrm{k}_{\mathfrak{X}}(G)=\mathrm{k}_{\mathfrak{X}}(\operatorname{im} \phi)$holds if and only if $\mathrm{k}_{\mathfrak{X}}(\ker \phi)=1$,which in turn is equivalent to the fact that the composition factors of the kernel of $\phi$ lie in an explicitly given list.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2022;86(6):79-100
79-100
Completeness of asymmetric products of harmonic functions and uniqueness of the solutionto the Lavrent'ev equation in inverse wave sounding problems
Abstract
We prove that the family of all pairwise products of regularharmonic functions in a domain $D \subset \mathbb{R}^3$ andNewtonian potentials of points located on a ray outside $D$ is complete in $L_2(D)$.This result is used for justification of uniqueness of a solution to the linear integral equationto which inverse problems of wave sounding in $\mathbb{R}^3$ are reduced.The corresponding inverse problems are shown to be uniquely solvable in spatially non-overdetermined settings wherethe dimension of the spatial data support coincides with that of the support of the sought-for function.Uniqueness theorems are used for establishingthat the axial symmetry of the input data for the inverse problemsunder consideration implies that of the solutions to these problems.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2022;86(6):101-122
101-122
Arithmetic of certain $\ell$-extensions ramified at three places. III
Abstract
Let $\ell$ be a regular odd prime, $K$ the $\ell$ th cyclotomic field and$K=k(\sqrt[\ell]{a} )$, where $a$ is a positive integer. Under theassumption that there are exactly three places ramified in the extension$K_\infty/k_\infty$, we study the $\ell$-component of the class group of thefield $K$. We prove that in the case $\ell>3$ there always is an unramifiedextension $\mathcal{N}/K$ such that $G(\mathcal{N}/K)\cong (\mathbbZ/\ell\mathbb Z)^2$ and all places over $\ell$ split completely in theextension $\mathcal{N}/K$. In the case $\ell=3$ we give a completedescription of the situation. Some other results are obtained.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2022;86(6):123-142
123-142
One class of quasilinear elliptic type equations with discontinuous nonlinearities
Abstract
In a bounded domain $\Omega\subset \mathbb{R}^n$, a class of quasilinear elliptic type boundary problems with parameter anddiscontinuous nonlinearity is studied.This class of problems includes the H. J. Kuiper conductor heating problem in a homogeneous electric field.The topological method is applied to verify the existence of a continuum of generalized positive solutionsfrom the Sobolev space $W_p^2(\Omega)$ ($p>n$) connecting $(0,0)$ with $\infty$in the space $\mathbb R\times C^{1,\alpha}(\overline\Omega)$, $\alpha\in (0,(p-n)/p)$. A sufficient conditionfor semiregularity of generalized solutions of this problem is given.The constraints on the discontinuous nonlinearityare relaxed in comparison with those used by H. J. Kuiper and K. C. Chang.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2022;86(6):143-160
143-160
Uniqueness sets of positive measure for the trigonometric system
Abstract
There exists a family $\mathcal{B}$ of one-to-one mappings $B \colon \mathbb{Z}\to\mathbb{Z}$ satisfying the condition $B(-n) \equiv -B(n)$ such that for each $B \in \mathcal{B}$ there exists a perfect uniqueness set of positive measure for the $B$-rearranged trigonometric system$\{\exp(iB(n)x)\}$. For a certain wider class of rearrangements of thetrigonometric system, the strengthened assertion holds from the Stechkin–Ul'yanovconjecture.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2022;86(6):161-186
161-186
Categories of weight modules for unrolled restricted quantum groups at roots of unity
Abstract
Motivated by connections to the singlet vertex operator algebra in the$\mathfrak{g}=\mathfrak{sl}_2$ case, we study the unrolledrestricted quantum group $\overline{U}_q^{ H}(\mathfrak{g})$ for any finitedimensional complex simple Lie algebra $\mathfrak{g}$ at arbitrary roots ofunity with a focus on its category of weight modules. We show that thebraid group action naturally extends to the unrolled quantum groups andthat the category of weight modules is a generically semi-simple ribboncategory (previously known only for odd roots) with trivial Mügercenter and self-dual projective modules.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2022;86(6):187-206
187-206
On the coprimeness relation from the viewpoint of monadic second-order logic
Abstract
Let $\mathfrak{C}$ denote the structure of the natural numbers with thecoprimeness relation. We prove that for each non-zero natural number $n$,if a $\Pi^1_n$-set of natural numbers is closed under automorphismsof $\mathfrak{C}$, then it is definable in $\mathfrak{C}$ by a monadic$\Pi^1_n$-formula of the signature of $\mathfrak{C}$ having exactly $n$ setquantifiers.On the other hand, we observe that even a much weaker version of this property fails for certain expansions of $\mathfrak{C}$.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2022;86(6):207-222
207-222
Approximative and structural properties of sets in asymmetric spaces
Abstract
Structural and approximative properties of sets implying their solarity are studied.It is shown that, in any finite-dimensionalpolyhedral space, each strict sun admits a continuous $\varepsilon$-selection for all $\varepsilon>0$ andthe metric projection onto it has cell-like values.In general asymmetric spaces, sufficient conditions for solarity of Chebyshev setsare put forward.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2022;86(6):223-238
223-238