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Volume 85, Nº 6 (2021)
- Ano: 2021
- Artigos: 8
- URL: https://bakhtiniada.ru/1607-0046/issue/view/7553
Articles
The issue is dedicated to the memory of Petr Sergeevich Novikov and Sergei Ivanovich Adian
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(6):3-4
3-4
Weights of exact threshold functions
Resumo
We consider Boolean exact threshold functions defined by linear equations and, more generally, polynomials of degree $d$. We give upper and lower bounds on the maximum magnitude (absolute value) of the coefficients required to represent such functions. These bounds are very close. In the linear case in particular they are almost matching. This quantity is the same as the maximum magnitude of the integer coefficients of linear equations required to express every possible intersection of a hyperplane in $\mathbb R^n$ and the Boolean cube $\{0,1\}^n$ or, in the general case, intersections of hypersurfaces of degree $d$ in $\mathbb R^n$ and $\{0,1\}^n$. In the process we construct new families of ill-conditioned matrices. We further stratify the problem (in the linear case) in terms of the dimension $k$ of the affine subspace spanned by the solutions and give upper and lower bounds in this case as well. There is a substantial gap between these bounds, a challenge for future work.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(6):5-26
5-26
On a spectral sequence for the action of the Torelli group of genus $3$ on the complex of cycles
Resumo
The Torelli group of a closed oriented surface $S_g$ of genus $g$ is the subgroup$\mathcal{I}_g$ of the mapping class group $\operatorname{Mod}(S_g)$ consisting ofall mapping classes that act trivially on the homology of $S_g$. One of the most intriguingopen problems concerning Torelli groups is the question of whether the group $\mathcal{I}_3$is finitely presented. A possible approach to this problem relies on the study of the secondhomology group of $\mathcal{I}_3$ using the spectral sequence $E^r_{p,q}$ for the actionof $\mathcal{I}_3$ on the complex of cycles. In this paper we obtain evidence forthe conjecture that $H_2(\mathcal{I}_3;\mathbb{Z})$ is not finitely generated and hence$\mathcal{I}_3$ is not finitely presented. Namely, we prove that the term $E^3_{0,2}$ ofthe spectral sequence is not finitely generated, that is, the group $E^1_{0,2}$ remainsinfinitely generated after taking quotients by the images of the differentials $d^1$ and $d^2$.Proving that it remains infinitely generated after taking the quotient by the imageof $d^3$ would complete the proof that $\mathcal{I}_3$ is not finitely presented.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(6):27-103
27-103
Finitely generated subgroups of branch groups and subdirect products of just infinite groups
Resumo
The aim of this paper is to describe the structure of finitely generated subgroups of a familyof branch groups containing the first Grigorchuk group and the Gupta–Sidki $3$-group. We thenuse this to show that all the groups in this family are subgroup separable (LERF).These results are obtained as a corollary of a more general structural statement on subdirectproducts of just infinite groups.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(6):104-125
104-125
Finitely presented nilsemigroups: complexes with the property of uniform ellipticity
Resumo
This paper is the first in a series of three devoted to constructing a finitely presented infinite nilsemigroup satisfying the identity $x^9=0$. This solves a problem of Lev Shevrin and Mark Sapir.In this first part we obtain a sequence of complexes formed of squares ($4$-cycles) having the following geometric properties.1) Complexes are uniformly elliptic. A space is said to be uniformly elliptic if there is a constant $\lambda>0$ such that in the set of shortest paths of length $D$ connecting points $A$ and $B$ there are two paths such that the distance between them is at most $\lambda D$. In this case, the distance between paths with the same beginning and end is defined as the maximal distance between the corresponding points. 2) Complexes are nested.A complex of level $n+1$ is obtained from a complex of level $n$ by adding several vertices and edges according to certain rules.3) Paths admit local transformations. Assume that we can transform paths by replacing a path along two sides of a minimal square by the path along the other two sides. Two shortest paths with the same ends can be transformed into each other locally if these ends are vertices of a square in the embedded complex.The geometric properties of the sequence of complexes will be further used to define finitely presented semigroups.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(6):126-163
126-163
Models of set theory in which the separation theorem fails
Resumo
We use a finite-support product of Jensen-minimal forcings to define a model of set theoryin which the separation theorem fails for the projective classes $\mathbf{\Sigma}^1_n$ and$\mathbf{\Pi}^1_n$, for a given $n\ge3$.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(6):164-204
164-204
The Diophantine problem in the classical matrix groups
Resumo
In this paper we study the Diophantine problem in the classical matrix groups $\mathrm{GL}_n(R)$, $\mathrm{SL}_n(R)$, $\mathrm{T}_n(R)$ and $\mathrm{UT}_n(R)$, $n \geq 3$, over an associative ring $R$ with identity. We show that if $G_n(R)$ is one of these groups, then the Diophantine problem in $G_n(R)$ is polynomial-time equivalent (more precisely, Karp equivalent) to the Diophantine problem in $R$. When $G_n(R)=\mathrm{SL}_n(R)$ we assume that $R$ is commutative. Similar results hold for $\mathrm{PGL}_n(R)$ and $\mathrm{PSL}_n(R)$ provided $R$ has no zero divisors (for $\mathrm{PGL}_n(R)$ the ring $R$ is not assumed to be commutative).
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(6):205-244
205-244
Lattice of definability (of reducts) for integers with successor
Resumo
In this paper the lattice of definability for integers with a successor (the relation $y = x + 1$) is described. The lattice, whose elements are also knows as reducts, consists of three(naturally described) infinite series of relations.The proof uses a version of the Svenonius theoremfor structures of special form.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(6):245-258
245-258