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Realization of Fomenko-Zieschang invariants in closed symplectic manifolds with contact singularities Sb. Math. (IF 0.8) Pub Date : 2022-04-01 D. B. Zot’ev, V. I. Sidel’nikov
The topological bifurcations of Liouville foliations on invariant -manifolds that are induced by attaching toric -handles are investigated. It is shown that each marked molecule (Fomenko-Zieschang invariant) can be realized on an invariant submanifold of a closed symplectic manifold with contact singularities which is obtained by attaching toric -handles sequentially to a set of symplectic manifolds
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Existence of solutions of nonlinear elliptic equations with measure data in Musielak-Orlicz spaces Sb. Math. (IF 0.8) Pub Date : 2022-04-01 A. P. Kashnikova, L. M. Kozhevnikova
A second-order quasilinear elliptic equation with a measure of special form on the right-hand side is considered. Restrictions on the structure of the equation are imposed in terms of a generalized -function such that the conjugate function obeys the -condition and the corresponding Musielak-Orlicz space is not necessarily reflexive. In an arbitrary domain satisfying the segment property, the existence
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How many roots of a system of random Laurent polynomials are real? Sb. Math. (IF 0.8) Pub Date : 2022-04-01 B. Ya. Kazarnovskii
We say that a zero of a Laurent polynomial that lies on the unit circle with centre is real. We also say that a Laurent polynomial that is real on this circle is real. In contrast with ordinary polynomials, it is known that for random real Laurent polynomials of increasing degree the average proportion of real roots tends to rather than to . We show that this phenomenon of the asymptotically nonvanishing
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Extremal functional -interpolation on an arbitrary mesh on the real axis Sb. Math. (IF 0.8) Pub Date : 2022-04-01 Yu. N. Subbotin, V. T. Shevaldin
The Golomb-de Boor problem of extremal interpolation of infinite real sequences with smallest -norm of the th derivative of the interpolant, , on an arbitrary mesh on the real axis is studied under constraints on the norms of the corresponding divided differences. For this smallest norm, lower estimates are obtained for any in terms of -splines. For the second derivative, this quantity is estimated
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Configuration spaces of hinged mechanisms, and their projections Sb. Math. (IF 0.8) Pub Date : 2022-04-01 M. D. Kovalev
Our subject is the geometry of planar hinged mechanisms. The article contains a formalization of basic concepts of the theory of hinged-lever constructions, as well as some information from real algebraic geometry needed for their study. We consider mechanisms with variable number of degrees of freedom and mechanisms that have more than one degree of freedom but each hinge of which moves with one degree
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Time minimization problem on the group of motions of a plane with admissible control in a half-disc Sb. Math. (IF 0.8) Pub Date : 2022-04-01 A. P. Mashtakov
The time minimization problem with admissible control in a half-disc is considered on the group of motions of a plane. The control system under study provides a model of a car on the plane that can move forwards or rotate in place. Optimal trajectories of such a system are used to detect salient curves in image analysis. In particular, in medical image analysis such trajectories are used for tracking
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Optimal recovery in weighted spaces with homogeneous weights Sb. Math. (IF 0.8) Pub Date : 2022-03-01 K. Yu. Osipenko
The paper concerns problems of the recovery of operators from noisy information in weighted -spaces with homogeneous weights. A number of general theorems are proved and applied to problems of the recovery of differential operators from a noisy Fourier transform. In particular, optimal methods are obtained for the recovery of powers of the Laplace operator from a noisy Fourier transform in the -metric
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On the problem of periodicity of continued fraction expansions of for cubic polynomials over algebraic number fields Sb. Math. (IF 0.8) Pub Date : 2022-03-01 V. P. Platonov, V. S. Zhgoon, M. M. Petrunin
We obtain a complete description of the fields that are extensions of of degree at most and the cubic polynomials such that the expansion of into a continued fraction in the field of formal power series is periodic. We prove a finiteness theorem for cubic polynomials with a periodic expansion of for extensions of of degree at most . We obtain a description of the periodic elements for the cubic polynomials
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A Chisini Theorem for almost generic covers of the projective plane Sb. Math. (IF 0.8) Pub Date : 2022-03-01 Vik. S. Kulikov
Results related to Chisini’s Conjecture and contained in (Izv. Math. 63:6 (1999), 1139–1170) and (Izv. Math. 65:1 (2001), 71–74) are extended to the case of almost generic covers of the projective plane.Bibliography: 11 titles.
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On the absence of global solutions of a system of ordinary differential equations Sb. Math. (IF 0.8) Pub Date : 2022-03-01 A. A. Kon’kov
Conditions for the absence of global solutions of a system of nonlinear ordinary differential equations are found. Examples showing that these conditions are sharp are given.Bibliography: 12 titles.
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On the cohomology rings of partially projective quaternionic Stiefel manifolds Sb. Math. (IF 0.8) Pub Date : 2022-03-01 G. E. Zhubanov, F. Yu. Popelenskii
The quaternionic Stiefel manifold is the total space of a fibre bundle over the corresponding Grassmannian . The group acts freely on the fibres of this bundle. The quotient space is called the quaternionic projective Stiefel manifold. Its real and complex analogues were actively studied earlier by a number of authors. A finite group acting freely on the three-dimensional sphere also acts freely and
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Lyapunov instability of stationary flows of a polymeric fluid in a channel with perforated walls Sb. Math. (IF 0.8) Pub Date : 2022-03-01 A. M. Blokhin, D. L. Tkachev
The rheological Pokrovskii-Vinogradov model for flows of solutions or melts of an incompressible viscoelastic polymeric medium is studied in the case of flows in an infinite planar channel with perforated walls. The linear Lyapunov instability is proved for the base solution with constant flow rate in the class of perturbations periodic in the variable varying along the channel wall.Bibliography: 14
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Bifurcations changing the homotopy type of the closure of an invariant saddle manifold of a surface diffeomorphism Sb. Math. (IF 0.8) Pub Date : 2022-03-01 E. V. Nozdrinova, O. V. Pochinka
It is well known from the homotopy theory of surfaces that an ambient isotopy does not change the homotopy type of a closed curve. Using the language of dynamical systems, this means that an arc in the space of diffeomorphisms that joins two isotopic diffeomorphisms with invariant closed curves in distinct homotopy classes must go through bifurcations. A scenario is described which changes the homotopy
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Values of the weight system on a family of graphs that are not the intersection graphs of chord diagrams Sb. Math. (IF 0.8) Pub Date : 2022-02-01 P. A. Filippova
The Chmutov-Lando theorem claims that the value of a weight system (a function on the chord diagrams that satisfies the four-term Vassiliev relations) corresponding to the Lie algebra depends only on the intersection graph of the chord diagram.We compute the values of the weight system at the graphs in several infinite series, which are the joins of a graph with a small number of vertices and a discrete
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Solarity and connectedness of sets in the space and in finite-dimensional polyhedral spaces Sb. Math. (IF 0.8) Pub Date : 2022-02-01 I. G. Tsar’kov
Generalized -piecewise functions constructed from given monotone path-connected boundedly compact subsets of the space are studied. They are shown to be monotone path-connected suns. In finite-dimensional polyhedral spaces, luminosity points of sets admitting a lower semicontinuous selection of the metric projection operator are investigated. An example of a non- -connected sun in a four-dimensional
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A hyperbolicity criterion for a class of diffeomorphisms of an infinite-dimensional torus Sb. Math. (IF 0.8) Pub Date : 2022-02-01 S. D. Glyzin, A. Yu. Kolesov
On an infinite-dimensional torus , where is an infinite-dimensional real Banach space and is an abstract integer lattice, a special class of diffeomorphisms is considered. It consists of the maps equal to sums of invertible bounded linear operators preserving and -smooth periodic additives. Necessary and sufficient conditions ensuring that such maps are hyperbolic (that is, are Anosov diffeomorphisms)
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Vinogradov’s sieve and an estimate for an incomplete Kloosterman sum Sb. Math. (IF 0.8) Pub Date : 2022-02-01 M. A. Korolev
We refine a bound for a short Kloosterman sum with a prime modulus using the so-called Vinogradov sieve. The number of terms in the sum can be less than an arbitrarily small fixed power of .Bibliography: 26 titles.
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Topological classification of billiards bounded by confocal quadrics in three-dimensional Euclidean space Sb. Math. (IF 0.8) Pub Date : 2022-02-01 G. V. Belozerov
We study billiards on compact connected domains in bounded by a finite number of confocal quadrics meeting in dihedral angles equal to . Billiards in such domains are integrable due to having three first integrals in involution inside the domain. We introduce two equivalence relations: combinatorial equivalence of billiard domains determined by the structure of their boundaries, and weak equivalence
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A necessary and sufficient condition for the existence of simple closed geodesics on regular tetrahedra in spherical space Sb. Math. (IF 0.8) Pub Date : 2022-02-01 A. A. Borisenko
A necessary and sufficient condition is obtained for the existence of a simple closed geodesic of type on a regular tetrahedron in spherical space.Bibliography: 6 titles.
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Convergence of two-point Pad approximants to piecewise holomorphic functions Sb. Math. (IF 0.8) Pub Date : 2022-01-22 M. L. Yattselev
Let and be formal power series at the origin and infinity, and , , be the rational function that simultaneously interpolates at the origin with order and at infinity with order . When germs and represent multi-valued functions with finitely many branch points, it was shown by Buslaev that there exists a unique compact set in the complement of which the approximants converge in capacity to the approximated
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Global boundedness of functions of finite order that are bounded outside small sets Sb. Math. (IF 0.8) Pub Date : 2022-01-22 B. N. Khabibullin
We prove that subharmonic or holomorphic functions of finite order on the plane, in space, or on the unit disc or ball that are bounded above on a sequence of circles or spheres, or on a system of embedded discs or balls, outside some asymptotically small sets are bounded above throughout. Hence, subharmonic functions of finite order on the complex plane, entire or plurisubharmonic functions of finite
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Estimates for the volume of the zeros of a holomorphic function depending on a complex parameter Sb. Math. (IF 0.8) Pub Date : 2022-01-22 A. M. Kytmanov, A. Sadullaev
Given a holomorphic function , , , an estimate for the volume of the zero set is presented which holds uniformly in . Such estimates are quite useful in investigations of oscillatory integrals of the form as . Here is a so-called amplitude function and is a phase function. Bibliography: 9 titles.
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On optimal recovery of values of linear operators from information known with a stochastic error Sb. Math. (IF 0.8) Pub Date : 2022-01-22 K. Yu. Krivosheev
The optimal recovery of values of linear operators is considered for classes of elements the information on which is known with a stochastic error. Linear optimal recovery methods are constructed that, in general, do not use all the available information for the measurements. As a consequence, an optimal method is described for recovering a function from a finite set of its Fourier coefficients specified
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A probability estimate for the discrepancy of Korobov lattice points Sb. Math. (IF 0.8) Pub Date : 2022-01-22 A. A. Illarionov
Bykovskii (2002) obtained the best current upper estimate for the minimum discrepancy of the Korobov lattice points from the uniform distribution. We show that this estimate holds for almost all -dimensional Korobov lattices of nodes, where , and is a prime number. Bibliography: 14 titles.
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Orthogonality in nonseparable rearrangement-invariant spaces Sb. Math. (IF 0.8) Pub Date : 2022-01-22 S. V. Astashkin, E. M. Semenov
Let be a nonseparable rearrangement-invariant space and let be the closure of the space of bounded functions in . Elements of orthogonal to , that is, elements , , such that for each , are investigated. The set of orthogonal elements is characterized in the case when is a Marcinkiewicz or an Orlicz space. If an Orlicz space is considered with the Luxemburg norm, then the set is the algebraic sum of
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New moduli components of rank 2 bundles on projective space Sb. Math. (IF 0.8) Pub Date : 2022-01-22 C. Almeida, M. Jardim, A. S. Tikhomirov, S. A. Tikhomirov
We present a new family of monads whose cohomology is a stable rank 2 vector bundle on . We also study the irreducibility and smoothness together with a geometrical description of some of these families. These facts are used to construct a new infinite series of rational moduli components of stable rank 2 vector bundles with trivial determinant and growing second Chern class. We also prove that the
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Entropy of a unitary operator on [IMG align=ABSMIDDLE alt=$ L^2(\mathbb{T}^n)$]tex_sm_5018_img1[/IMG] Sb. Math. (IF 0.8) Pub Date : 2022-01-01 Kirill Aleksandrovich Afonin,Dmitrii Valer'evich Treschev
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Asymptotics of the sphere and front of a flat sub-Riemannian structure on the Martinet distribution Sb. Math. (IF 0.8) Pub Date : 2022-01-01 Il'ya Aleksandrovich Bogaevsky
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On metrical properties of [IMG align=ABSMIDDLE alt=$ C$]tex_sm_5017_img1[/IMG]-capacities associated to solutions of second-order strongly elliptic equations in [IMG align=ABSMIDDLE alt=$ \mathbb R^2$]tex_sm_5017_img2[/IMG] Sb. Math. (IF 0.8) Pub Date : 2022-01-01 Petr Vladimirovich Paramonov
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On the universality for zeta-functions of some certain cusp forms Sb. Math. (IF 0.8) Pub Date : 2022-01-01 Antanas P Laurinčikas
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Hardy-Littlewood-Sobolev inequality for [IMG align=ABSMIDDLE alt=$ p=1$]tex_sm_5015_img1[/IMG] Sb. Math. (IF 0.8) Pub Date : 2022-01-01 Dmitry Mikhailovich Stolyarov
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Integrals of the difference of subharmonic functions with respect to measures and the Nevanlinna characteristic Sb. Math. (IF 0.8) Pub Date : 2022-01-01 Bulat Nurmievich Khabibullin
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On tensor fractions and tensor products in the category of stereotype spaces Sb. Math. (IF 0.8) Pub Date : 2022-01-01 Sergei Saidmuzafarovich Akbarov
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Canonical geometrization of orientable [IMG align=ABSMIDDLE alt=$ 3$]tex_sm_5016_img1[/IMG]-manifolds defined by vector-colourings of [IMG align=ABSMIDDLE alt=$ 3$]tex_sm_5016_img1[/IMG]-polytopes Sb. Math. (IF 0.8) Pub Date : 2022-01-01 Nikolai Yur'evich Erokhovets
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[IMG align=ABSMIDDLE alt=$ p$]tex_sm_5014_img1[/IMG]-convexity functor for [IMG align=ABSMIDDLE alt=$ L_p(X)$]tex_sm_5014_img2[/IMG]-spaces Sb. Math. (IF 0.8) Pub Date : 2022-01-01 Nina Vladimirovna Volosova
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Geormtry of the Gromov-Hausdorff Distance on the Class of all Metric Spaces Sb. Math. (IF 0.8) Pub Date : 2022-01-01 Stanislav Igorevich Borzov,Alexandr Olegovich Ivanov,Alexey Avgustinovich Tuzhilin
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Central extensions and Riemann-Roch theorem on algebraic surfaces Sb. Math. (IF 0.8) Pub Date : 2022-01-01 Denis Vasilievich Osipov
We study canonical central extensions of the general linear group over the ring of adeles on a smooth projective algebraic surface X by means of the group of integers. By these central extensions and adelic transition matrices of a rank n locally free sheaf of OX -modules we obtain the local (adelic) decomposition for the difference of Euler characteristics of this sheaf and the sheaf On X . Two various
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Strong convexity of reachable sets of linear systems Sb. Math. (IF 0.8) Pub Date : 2022-01-01 Maxim Viktorovich Balashov
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Self-affine attractors and tiles Sb. Math. (IF 0.8) Pub Date : 2022-01-01 Tatyana Ivanovna Zaitseva,Vladimir Yur'evich Protasov
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On singular log Calabi-Yau compactifications of Landau-Ginzburg models Sb. Math. (IF 0.8) Pub Date : 2022-01-01 V. V. Przyjalkowski
We consider the procedure that constructs log Calabi-Yau compactifications of weak Landau-Ginzburg models of Fano varieties. We apply it to del Pezzo surfaces and coverings of projective spaces of index . For coverings of degree greater than the log Calabi-Yau compactification is singular; moreover, no smooth projective log Calabi-Yau compactification exists. We also prove, in the cases under consideration
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On the local and boundary behaviour of inverse maps on Riemannian manifolds Sb. Math. (IF 0.8) Pub Date : 2022-01-01 D. P. Ilyutko, E. A. Sevost’yanov
Results on the local behaviour of maps between Riemannian manifolds such that their inverses satisfy upper bounds on the distortion of the moduli of families of curves are obtained. For families of such maps theorems on their equicontinuity at interior points and boundary points of the domain are established.Bibliography: 30 titles.
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Nonlocal balance equations with parameters in the space of signed measures Sb. Math. (IF 0.8) Pub Date : 2022-01-01 N. I. Pogodaev, M. V. Staritsyn
A parametric family of nonlocal balance equations in the space of signed measures is studied. Under assumptions that cover a number of known conceptual models we establish the existence of the solution, its uniqueness and continuous dependence on the parameter and the initial distribution. Several corollaries of this theorem, which are useful for control theory, are discussed. In particular, this theorem
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More about sparse halves in triangle-free graphs Sb. Math. (IF 0.8) Pub Date : 2022-01-01 A. A. Razborov
One of Erdős’s conjectures states that every triangle-free graph on vertices has an induced subgraph on vertices with at most edges. We report several partial results towards this conjecture. In particular, we establish the new bound on the number of edges in the general case. We completely prove the conjecture for graphs of girth , for graphs with independence number and for strongly regular graphs
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Global and semilocal theorems on implicit and inverse functions in Banach spaces Sb. Math. (IF 0.8) Pub Date : 2022-01-01 A. V. Arutyunov, S. E. Zhukovskiy
We consider continuous mappings between two Banach spaces that depend on a parameter with values in a topological space. These mappings are assumed to be continuously differentiable for each value of the parameter. Under normality (regularity) assumptions of the mappings under consideration, we obtain sufficient conditions for the existence of global and semilocal implicit functions. A priori estimates
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A Littlewood-Paley-Rubio de Francia inequality for bounded Vilenkin systems Sb. Math. (IF 0.8) Pub Date : 2021-12-14 A. S. Tselishchev
Rubio de Francia proved a one-sided Littlewood-Paley inequality for the square function constructed from an arbitrary system of disjoint intervals. Later, Osipov proved a similar inequality for Walsh systems. We prove a similar inequality for more general Vilenkin systems. Bibliography: 11 titles.
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Slide polynomials and subword complexes Sb. Math. (IF 0.8) Pub Date : 2021-12-14 E. Yu. Smirnov, A. A. Tutubalina
Subword complexes were defined by Knutson and Miller in 2004 to describe Grbner degenerations of matrix Schubert varieties. Subword complexes of a certain type are called pipe dream complexes. The facets of such a complex are indexed by pipe dreams, or, equivalently, by monomials in the corresponding Schubert polynomial. In 2017 Assaf and Searles defined a basis of slide polynomials, generalizing Stanley
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Asymptotics of the scattering operator for the wave equation in a singularly perturbed domain Sb. Math. (IF 0.8) Pub Date : 2021-12-14 D. V. Korikov
A family of Cauchy-Dirichlet problems for the wave equations in unbounded domains is considered (here is a small parameter); a scattering operator is associated with each domain . For the boundaries of the are smooth, while the boundary of the limit domain contains a conical point. The asymptotics of as is determined. Bibliography: 11 titles.
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The regularized asymptotics of a solution of the Cauchy problem in the presence of a weak turning point of the limit operator Sb. Math. (IF 0.8) Pub Date : 2021-12-14 A. G. Eliseev
An asymptotic solution of the linear Cauchy problem in the presence of a ‘weak’ turning point of the limit operator is built using Lomov’s regularization method. The major singularities of the problem are written out in an explicit form. Estimates are given with respect to , which characterise the behaviour of the singularities as . The asymptotic convergence of the regularized series is proved. The
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The degrees of maps between -connected -dimensional manifolds or Poincar complexes and their applications Sb. Math. (IF 0.8) Pub Date : 2021-12-14 J. Grbić, A. Vučić
In this paper, using homotopy theoretical methods we study the degrees of maps between -connected -dimensional Poincar complexes. Necessary and sufficient algebraic conditions for the existence of mapping degrees between such Poincar complexes are established. These conditions allow us, up to homotopy, to construct explicitly all maps with a given degree. As an application of mapping degrees, we consider
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Stationary points of the Minkowski function Sb. Math. (IF 0.8) Pub Date : 2021-12-14 D. R. Gayfulin, I. D. Kan
A new theorem on the derivative of the Minkowski function is proved. Bibliography: 11 titles.
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Uniqueness theorems for simple trigonometric series with application to multiple series Sb. Math. (IF 0.8) Pub Date : 2021-12-01 G. G. Gevorkyan
Abstract For simple trigonometric series it is shown, in particular, that if the trigonometric series is Riemann summable in measure to an integrable function and if the Riemann majorant is finite everywhere except possibly on a countable set, then this series is the Fourier series of the function . Uniqueness theorems for multiple trigonometric series are obtained on the basis of this result. Bibliography:
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Topological type of isoenergy surfaces of billiard books Sb. Math. (IF 0.8) Pub Date : 2021-12-01 V. V. Vedyushkina
Abstract The homeomorphism class of the isoenergy surface of a billiard book, of low complexity and not necessarily integrable, is determined using methods of low-dimensional topology. In particular, a series of billiard books is constructed that realize isoenergy 3-surfaces homeomorphic to the connected sum of a number of lens spaces and direct products . The Fomenko-Zieschang invariants, which classify
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Uniform approximation of functions by solutions of strongly elliptic equations of second order on compact subsets of Sb. Math. (IF 0.8) Pub Date : 2021-12-01 P. V. Paramonov
Abstract Criteria for the uniform approximation of functions by solutions of second-order strongly elliptic equations on compact subsets of are obtained using the method of reduction to similar problems in , which were previously investigated by Mazalov. A number of metric properties of the capacities used are established. Bibliography: 16 titles.
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Generalization of the Artin-Hasse logarithm for the Milnor -groups of -rings Sb. Math. (IF 0.8) Pub Date : 2021-12-01 D. N. Tyurin
Abstract Let be a -adically complete ring equipped with a -structure. We construct a functorial group homomorphism from the Milnor -group to the quotient of the -adic completion of the module of differential forms . This homomorphism is a -adic analogue of the Bloch map defined for the relative Milnor -groups of nilpotent extensions of rings of nilpotency degree for which the number is invertible.
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Orbit spaces for torus actions on Hessenberg varieties Sb. Math. (IF 0.8) Pub Date : 2021-12-01 V. V. Cherepanov
Abstract In this paper we study effective actions of the compact torus on smooth compact manifolds of even dimension with isolated fixed points. It is proved that under certain conditions on the weight vectors of the tangent representation, the orbit space of such an action is a manifold with corners. In the case of Hamiltonian actions, the orbit space is homotopy equivalent to , the complement to
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The polynomial Hermite-Padé -system for meromorphic functions on a compact Riemann surface Sb. Math. (IF 0.8) Pub Date : 2021-12-01 A. V. Komlov
Abstract Given a tuple of germs of arbitrary analytic functions at a fixed point, we introduce the polynomial Hermite-Padé -system, which includes the Hermite-Padé polynomials of types I and II. In the generic case we find the weak asymptotics of the polynomials of the Hermite-Padé -system constructed from the tuple of germs of functions that are meromorphic on an -sheeted compact Riemann surface
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The maximum tree of a random forest in the configuration graph Sb. Math. (IF 0.8) Pub Date : 2021-11-15 Yu. L. Pavlov
Galton-Watson random forests with a given number of root trees and a known number of nonroot vertices are investigated. The distribution of the number of direct offspring of each particle in the forest- generating process is assumed to have infinite variance. Branching processes of this kind are used successfully to study configuration graphs aimed at simulating the structure and development dynamics
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Rigid germs of finite morphisms of smooth surfaces and rational Belyi pairs Sb. Math. (IF 0.8) Pub Date : 2021-11-15 Vik. S. Kulikov
In the paper “On rigid germs of finite morphisms of smooth surfaces” (Sb. Math., 211:10 (2020), 1354–1381), we defined a map from the set of equivalence classes of rigid germs of finite morphisms branched in germs of curves having singularity types onto the set of rational Belyi pairs , considered up to the action of . In this article the inverse images of this map are investigated in terms of monodromies
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A Viskovatov algorithm for Hermite-Pad polynomials Sb. Math. (IF 0.8) Pub Date : 2021-11-15 N. R. Ikonomov, S. P. Suetin
We propose and justify an algorithm for producing Hermite- Pad polynomials of type I for an arbitrary tuple of formal power series , , about the point () under the assumption that the series have a certain (‘general position’) nondegeneracy property. This algorithm is a straightforward extension of the classical Viskovatov algorithm for constructing Pad polynomials (for our algorithm coincides with
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On irregular Sasaki-Einstein metrics in dimension Sb. Math. (IF 0.8) Pub Date : 2021-11-15 H. S
We show that there are no irregular Sasaki-Einstein structures on rational homology 5-spheres. On the other hand, using -stability we prove the existence of continuous families of nontoric irregular Sasaki- Einstein structures on odd connected sums of . Bibliography: 30 titles.