Vol 89, No 2 (2025)

We consider algebras of finite subsets under the assumption that theoriginal algebra is an infinite groupoid.For linear spaces over fields of finite characteristic,we prove that the finite subsets algebra is algorithmically equivalent to the first-order arithmetic.We also generalize this result to arbitrary infinite Abelian groups.As a corollary, for many classes of original algebrassuch as Abelian groups, arbitrary groups, monoids, semigroups, and general groupoids,if we consider the corresponding class of all algebras of finite subsets, it is shownthat the theory of the last class has degree of undecidabilitynot smaller than the first-order arithmetic.This also proves thatthe theories of such classes have no recursive axiomatization.For Abelian groups of finite exponent,we show that the theories of the subalgebra lattices are algorithmically undecidableand have no recursive axiomatization;also, for the class of such lattices for groups, monoids, or semigroups,we show that the theory of this class is also undecidable and has no recursive axiomatization.

We prove that any unconditional set in $\mathbb{R}^N$ invariant under cyclicshifts of coordinates is rigid in $\ell_q^N$, $1\le q\le 2$, that is, it cannot be well approximated by linear spaces of dimension essentially smaller than $N$. We apply E. D. Gluskin's approach to the setting of averaged Kolmogorov widths of unconditional random vectors or vectors of independent mean zero random variables, and prove their rigidity. These results are obtained using a general lower bound for the averaged Kolmogorov width via weak moments of biorthogonal random vector. This paper continues the study of the rigidity initiated by the first author.We also provide several corollaries including new bounds for Kolmogorov widths of mixed norm balls.

Let $R$ be an unramified regular local ring of mixed characteristic, $D$ an Azumaya $R$-algebra, $K$ the fraction field of $R$, $\operatorname{Nrd}\colon D^{\times} \to R^{\times}$ the reduced norm homomorphism. Let $a \in R^{\times}$ be a unit. Suppose the equation $\operatorname{Nrd}=a$ has a solution over $K$, then it has a solution over $R$.Particularly, we prove the following. Let $R$ be as above and $a$, $b$, $c$ be units in $R$. Consider the equation $T^2_1-aT^2_2-bT^2_3+abT^2_4=c$. If it has a solution over $K$, then it has a solution over $R$.Similar results are proved for regular local rings, which are geometrically regular over a discrete valuation ring. These results extend result provedin [23] to the mixed characteristic case. 

This study focuses on a specific type of non-linear second-order integral equations defined over a considerable interval. We propose a new numerical method, the TLD, in three phases: transformation of the equation, linearizationvia the Newton–Kantorovich method, and discretization withthe Nyström technique. We theoretically define the convergence conditions and illustrate the accuracy of our method on several practical examples.

We discuss global tensor invariants of a rigid body motion in the cases of Euler, Lagrange and Kovalevskaya.These invariants are obtained by substituting tensor fields with componentscubic in the variables into the invariance equation and solving the resulting algebraic equations using computer algebra systems.