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Vol 214, No 5 (2023)

Year: 2023
Articles: 6
URL: https://journal-vniispk.ru/0368-8666/issue/view/7502

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On Arf-invariants in the codimension $1$ in a Wall group of the dihedral group

Akhmet'ev P.M., Muranov Y.V.

Abstract

An element $x$ is specified in the Wall group $L_{3} (D_{3})$ of the dihedral group of order $8$ with trivial orientation character, such that $x$ is an element of the third type in the sense of Kharshiladze with respect to any system of one-sided submanifolds of codimension $1$ for which the splitting obstruction group along the first submanifold is isomorphic to $L N_{1} (Z / 2 \oplus Z / 2 \to D_{3})$ . The element $x$ is not realisable as an obstruction to surgery on a closed $P L$ -manifold. It is also proved that the unique nontrivial element of the group $L N_{3} (Z / 2 \oplus Z / 2 \to D_{3}^{-})$ can be detected using the Hasse-Witt $W h_{2}$ -torsion.

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Matematicheskii Sbornik. 2023;214(5):3-17

3-17

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A refinement of Heath-Browns theorem on quadratic forms

Vlăduţ S.G., Dymov A.V., Kuksin S.B., Maiocchi A.

Abstract

In his paper from 1996 on quadratic forms Heath-Brown developed a version of the circle method to count points in the intersection of an unbounded quadric with a lattice of small period, when each point is assigned a weight, and approximated this quantity by the integral of the weight function against a measure on the quadric. The weight function is assumed to be -smooth and vanish near the singularity of the quadric. In our work we allow the weight function to be finitely smooth, not to vanish at the singularity and have an explicit decay at infinity.
The paper uses only elementary number theory and is available to readers with no number-theoretic background.

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Matematicheskii Sbornik. 2023;214(5):18-68

18-68

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The theorems Levinson's type and Dynkin's problems

Gaisin A.M., Gaisin R.A.

Abstract

Questions relating to theorems of Levinson-Sjöberg-Wolf type in complex and harmonic analysis are explored. The well-known Dyn'kin problem of effective estimation of the growth majorant of an analytic function in a neighbourhood of its set of singularities is discussed, together with the problem, dual to it in certain sense, on the rate of convergence to zero of the extremal function in a nonquasianalytic Carleman class in a neighbourhood of a point at which all the derivatives of functions in this class vanish.
The first problem was solved by Matsaev and Sodin. Here the second Dyn'kin problem, going back to Bang, is fully solved. As an application, a sharp asymptotic estimate is given for the distance between the imaginary exponentials and the algebraic polynomials in a weighted space of continuous functions on the real line.

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Matematicheskii Sbornik. 2023;214(5):69-96

69-96

(Rus)

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Combinatorial invarian for gradient-like flows on connected summ of $\mathbb{S}^{n-1}\times \mathbb{S}^1$

Grines V.Z., Gurevich E.Y.

Abstract

We obtain necessary and sufficient conditions for the topological equivalence of gradient-like flows without heteroclinic intersections defined on the connected sum of a finite number of manifolds homeomorphic to , . For , this result extends substantially the class of manifolds such that structurally stable systems on these manifolds admit a topological classification.

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Matematicheskii Sbornik. 2023;214(5):97-127

97-127

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About some classes of almost Hermitian structures, which realized on $S^6$

Daurtseva N.A.

Abstract

Structures of cohomogeneity one on $S^{6}$ are under investigation. Examples of semi-Kähler and quasi-Kähler structures are constructed. Questions concerning the existence of almost Hermitian structures of cohomogeneity one on a round sphere are investigated.

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Matematicheskii Sbornik. 2023;214(5):128-139

128-139

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Slim exceptional sets of Waring-Goldbach problems involving squares and cubes of primes

Han X., Liu H.

Abstract

Let $p_{1}, p_{2}, \dots, p_{6}$ be prime numbers. First we show that, with at most $O (N^{1 / 12 + ε})$ exceptions, all even positive integers not exceeding $N$ can be represented in the form $p_{1}^{2} + p_{2}^{2} + p_{3}^{3} + p_{4}^{3} + p_{5}^{3} + p_{6}^{3}$ , which improves the previous result $O (N^{1 / 4 + ε})$ obtained by Y. H. Liu. Moreover, we also prove that, with at most $O (N^{5 / 12 + ε})$ exceptions, all even positive integers not exceeding $N$ can be represented in the form $p_{1}^{2} + p_{2}^{3} + p_{3}^{3} + p_{4}^{3} + p_{5}^{3} + p_{6}^{3}$ .

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Matematicheskii Sbornik. 2023;214(5):140-152

140-152

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