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Vol 216, No 10 (2025)
Fragments of arithmetic and cyclic proofs
Abstract
We define a new cyclic proof system for Peano arithmetic that that is simpler than existing one and can be adapted for an analysis of formal inference and for automatizing inductive proof search. We show how various traditional subsystems of Peano arithmetic defined by restricted forms of induction can be represented as fragments of the system proposed.
Matematicheskii Sbornik. 2025;216(10):3-28
3-28
Universal equivalence of general linear groups over local rings with 1/2
Abstract
It is proved that the universal equivalence of full linear groups of order strictly greater than 2 over local, not necessarily commutative rings with 1/2 is equivalent to the coincidence of their orders and the universal equivalence of the respective rings or the universal equivalence of one ring to the ring opposite to the other.
Matematicheskii Sbornik. 2025;216(10):29-41
29-41
Maximal Calderon–Zygmund operators and Weyl multipliers
Abstract
Let $T_k$ , $k=1,2,…,N$ , be a sequence of bounded operators on $L^p$ , $1, and $T^*(f)=\max_{1\le k\le N}|T_k(f)|$ . For some choices of $T_k$ it is of interest the problem of finding the optimal constant $c(N)$ for the bound
$$ \|T^*\|_{L^p\to L^p}\lesssim c(N) \max_{1\le k\le N}\|T_k\|_{L^p\to L^p}. $$
We consider this problem for Calderon–Zygmund operators. It was prove by first two authors that$c(N)\lesssim \log N$ when $T_k$ are general Calderon–Zygmund operators with uniformly bounded parameters. In this note we consider Calderon–Zygmund operators with kernels, having certain dyadic decomposition. We prove for such operators $c(N)\lesssim\sqrt{\log N}$ . Applying this bound, we prove that the sequence $\log n$ is an almost everywhere convergence Weyl multiplier for any rearranged dyadic block trigonometric polynomials.
$$ \|T^*\|_{L^p\to L^p}\lesssim c(N) \max_{1\le k\le N}\|T_k\|_{L^p\to L^p}. $$
We consider this problem for Calderon–Zygmund operators. It was prove by first two authors that
Matematicheskii Sbornik. 2025;216(10):42-61
42-61
Construction of polynomials in bi-involution for singular elements of the dual space of a Lie algebra
Abstract
A generalization of the well-known problem of he construction of complete full bi-involutive sets of polynomials on the conjugate space of a Lie algebra to the case of singular covectors is considered. A generalization of the Mishchenko–Fomenko method of argument shift to the case of singular covectors if proposed and sufficient conditions for the completeness of the resulting sets are found. Using this method, it is shown that complete bi-involutive sets of polynomials can be constructed for singular covectors in all reductive Lie algebras.
Matematicheskii Sbornik. 2025;216(10):62-76
62-76
Prismatic cohomology and de Rham–Witt forms
Abstract
For any prism $(A,d)$ , we construct an analogue of Fontaine's map $W_r(A/d) \to A/d\phi(d)\cdots\phi^{r-1}(d)$ . Subsequently, we define a canonical map from de Rham–Witt forms to prismatic cohomology in the perfect case and prove that it is an isomorphism. Using this result, we obtain an explicit description of the prismatic cohomology $H^i((S/A)_\Prism,\mathcal{O}_\Prism/d\phi(d)\cdots\phi^{n-1}(d))$, where $S$ is the $p$ -completion of a polynomial algebra over $A/d$ .
Matematicheskii Sbornik. 2025;216(10):77-100
77-100
Level surfaces of the first integral for a billiard system with cosine refraction
Abstract
A new integrable system in an ellipse is introduced: the domain bounded by an ellipse is partitioned into subdomains by arcs of confocal quadrics and each subdomain is filled by a medium with fixed constant coefficient of ‘optical’ density. In crossing an interface between media the trajectory obeys the ‘cosine law’ of refraction. It is shown that such systems have an additional first integral.
For two partitions of an ellipse into subdomains the level surfaces of the additional integral are examined in detail, as well as their bifurcations arising when going over critical values of the integral.
For two partitions of an ellipse into subdomains the level surfaces of the additional integral are examined in detail, as well as their bifurcations arising when going over critical values of the integral.
Matematicheskii Sbornik. 2025;216(10):101-158
101-158
Nonlinear growth of the Chebyshev norm of matrices under maximal cross approximation
Abstract
For the function $g(n)$ describing the maximal possible growth of the Chebyshev norms of maximal cross approximations of an $n\times n$ matrix, the inequality $4g(2k)\leqslant g(7k+3)$ is proved. The bound $g(n)\geqslant Cn^{\log_{7/2}4}$ is established on this basis.
Matematicheskii Sbornik. 2025;216(10):159-168
159-168
Letter to the editors
Matematicheskii Sbornik. 2025;216(10):169-170
169-170