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Vol 87, No 4 (2023)
Articles
3-46
The boundary behavior of $\mathcal Q_{p,q}$-homeomorphisms
Abstract
This article studies systematically the boundary correspondence problem for $\mathcal Q_{p,q}$-homeomorphisms. The presented example demonstrates a deformation of the Euclidean boundary with the weight function degenerating on the boundary.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(4):47-90
47-90
Variations of $v$-change of time in an optimal control problem with state and mixed constraints
Abstract
For a general optimal control problem with state and regular mixed constraints we propose a proof of the maximum principle by means of the so-called $v-$change of time $t \mapsto \tau,$ under which the original time becomes one more state variable subject to equation $dt/d\tau = v(\tau),$ while the additional control $v(\tau)\ge 0$ is piecewise constant and its values become arguments of the new problem.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(4):91-132
91-132
$SU$-linear operations in complex cobordism and the $c_1$-spherical bordism theory
Abstract
We study the $SU$-linear operations in complex cobordism and prove that they are generated by the well-known geometric operations $\partial_i$. For the theory $W$ of $c_1$-spherical bordism, we describe all $SU$-linear multiplications on $W$ and projections $MU \to W$. We also analyse complex orientations on $W$ and the corresponding formal group laws $F_W$. The relationship between the formal group laws $F_W$ and the coefficient ring $W_*$ of the $W$-theory was studied by Buchstaber in 1972. We extend his results by showing that for any $SU$-linear multiplication and orientation on $W$, the coefficients of the corresponding formal group law $F_W$ do not generate the ring $W_*$, unlike the situation with complex bordism.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(4):133-165
133-165
Unsolvability of the submonoid membership problem for a free nilpotent group of class $l\geq 2$ of a sufficiently large rank
Abstract
The paper gives an answer to the question of M. Lohrey and B. Steinberg about the solvability of the submonoid membership problem for a finitely generated nilpotent group. Namely, a finitely generated submonoid of a free nilpotent group of class $2$ of a sufficiently large rank $r$ is constructed, the membership problem for which is algorithmically unsolvable. This implies the existence of a submonoid with a similar property in any free nilpotent group of class $l \geq 2$ of rank $r$. The proof is based on the undecidability of Hilbert's 10th problem.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(4):166-185
166-185
On stabilization of solutions of semilinear parabolic second order equations on closed manifolds
Abstract
The work is devoted to existence, uniqueness, and stabilization of the weak solutions of one class of semilinear parabolic second order differential equations on closed manifolds. These equations are nonhomogeneous analogs of Kolmogorov–Petrovsky–Piskunov–Fisher equation and are of high importance from both applied and pure mathematical points of view.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(4):186-204
186-204
Continuous selections from multivalued maps and approximation in asymmetric and semi-linear spaces
Abstract
Michaels theorem on continuous selection from multivalued not necessarily convex maps is generalized.Classical approximation problems on cone spaces for symmetric and non-symmetric seminorms are considered.In particular, the conditions guaranteeing the existence of continuous selection for convex sets in non-symmetric spaces are studied.On a semilinear space of bounded convex sets with a Hausdorff semimetricity, the Chebyshev center problem is solved for bounded families of these sets.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(4):205-224
205-224