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Vol 217, No 1 (2026)
Limit theorems for functionals of a branching process in random environment starting with a large number of particles
Abstract
Assume given a sequence $Z^{(k)}=Ż_{i}^{(k)}, i=0,1,…\}$ , $k=1,2,…$ , of critical branching processes in random environment which are only different from one another by the size $k$ of the founding generation. Assume that the step of the associated random walk belongs to the domain of attraction of a stable law. In the case $k=k(n)$ , where $n$ is a positive integer parameter and $k(n)$ grows with $n$ in a certain special way, limit theorems as $n\to\infty$ are established for a process with continuous time constructed from $Z^{(k(n))}$ and for the logarithm of this process. In addition, limit theorems are proved for the moment of degeneration of the process $Z^{(k(n))}$ , the maximum of this process, and the total number of particles.
Matematicheskii Sbornik. 2026;217(1):3-28
3-28
29-53
Modular values of continuants with fixed prefixes and endings
Abstract
Consider the set of finite words in a finite alphabet $\mathbf{A}\subseteq\mathbb{N}$ . Add a prefix $V$ and an ending $W$ , which are some fixed finite words in the alphabet $\mathbb{N}$ , to each word. We interpret the resulting words as the expansions in finite continued fractions of some rational numbers in the interval $(0,1)$ . Next consider the irreducible denominators of these rational numbers; we denote the set of those denominators that do not exceed some quantity $N\in \mathbb{N}$ (which is an increasing parameter) by $\mathfrak{D}^{N}_{\mathbf{A},V,W}$ . We prove that under certain conditions on $\mathbf{A}$ , $V$ and $W$ , for each prime number $Q$ proportional to a fixed fractional power of $N$ the set $\mathfrak{D}^{N}_{\mathbf{A},V,W}$ contains almost all possible residues modulo $Q$ , and the remainder in this asymptotic formula involves a power reduction with respect to $Q$ .
Matematicheskii Sbornik. 2026;217(1):54-88
54-88
Critical vector spaces and extremal $L$ -varieties
Abstract
An example of an almost commutative, almost Engel and almost finitely based $L$ -variety is presented that is generated by a multiplicative vector space distinct from its enveloping algebra. Also, an example of a critical multiplicative vector space is constructed that is not a linear algebra.
Matematicheskii Sbornik. 2026;217(1):89-97
89-97
Spectra and joint dynamics of Poisson suspensions over rank-one automorphisms
Abstract
For each integer $n>1$ a unitary operator of dynamical origin is found such that its $n$ th tensor power has a singular spectrum, but the spectrum of the $(n+1)$ st power is absolutely continuous. For any sequences $p(n)$ and $q(n)$ , provided that $ p(n+1)- p(n) \to+\infty$ and $ q(n+1)- q(n)\to +\infty$ , there exist a set $C$ and automorphisms $S$ and $T$ with simple singular spectra such that the sequence $ \sum_{n=1}^{N} \mu(S^{ p(n)}C\cap T^{ q(n)}C)/N$ is divergent. In the class of Poisson suspensions with zero entropy there exist mixing automorphisms $S$ and $T$ such that for some set $D$ of positive measure, $S^nD\cap T^nD=\varnothing$ for each $n>0$ .
Matematicheskii Sbornik. 2026;217(1):98-113
98-113
On nonexistence of negative weight derivations on moduli algebras: Yau's conjecture
Abstract
Let $A=\mathbb{F}[x_1,…,x_n]/(f_1,…,f_n)$ be a graded complete intersection Artinian algebra where $\mathbb{F}$ is a field of characteristic zero. The grading on $A$ induces a natural grading on $\operatorname{Der}_{\mathbb{F}}(A)$ . Halperin proposed a famous conjecture: $\operatorname{Der}_{\mathbb{F}}(A)_{<0}=0$ , which implies the collapsing of the Serre spectral sequence for an orientable fibration with fibre an elliptic space with no cohomology in odd degrees. In the context of singularity theory the second author proposed the same conjecture in the special case when $f_i=\partial f/\partial x_i$ for a single polynomial $f$ .
H. Chen, the second author and Zuo [5] proved Halperin's conjecture assuming that the degrees of the$f_i$ are bounded below by a constant depending on the number $n$ of variables and the degrees of variables. In this paper, in the special case when $f_i=\partial f/\partial x_i$ for a single polynomial $f$ , we refine their result by giving a better bound, which is independent of $n$ .
H. Chen, the second author and Zuo [5] proved Halperin's conjecture assuming that the degrees of the
Matematicheskii Sbornik. 2026;217(1):114-138
114-138