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Working paper

Given two fields k and F, a k-vector space V, an integer r≥1, we study the structure of the F-representation of the projective group PGL(V) in the F-vector space of formal finite linear combinations with coefficients in F of r-dimensional vector subspaces of V.
This gives a series of natural examples of irreducible infinite-dimensional representations of projective groups. These representations are non-smooth when k is a local field.

Added: Dec 5, 2018

Working paper

We develop the formal analogue of the Morse theory for a pair of commuting gradient-like vector fields. The resulting algebraic formalism turns out to be very similar to the algebra of the infrared of Gaiotto, Moore and Witten (see [GMW], [KKS]): from a manifold M with the pair of gradient-like commuting vector fields, subject to some general position conditions we construct an L_∞-algebra and Maurer-Cartan element in it.
We also provide Morse-theoretic examples for the algebra of the infrared data.

Added: Dec 1, 2018

Working paper

In the first section of this work we introduce 4-dimensional Power Geometry for second-order ODEs of a polynomial form. In the next five sections we apply this construction to the first five Painlev ́e equations.

Added: Mar 28, 2015

Working paper

We list all finite abelian groups which act effectively on smooth cubic fourfolds.

Added: Nov 19, 2013

Working paper

We obtain a partial solution of the problem on the growth of the norms of exponential functions with a continuous phase in the Wiener algebra. The problem was posed by J.-P. Kahane at the International Congress of Mathematicians in Stockholm in 1962. He conjectured that (for a nonlinear phase) one can not achieve the growth slower than the logarithm of the frequency. Though the conjecture is still not confirmed, the author obtained first nontrivial results.

Added: Apr 12, 2012

Working paper

We prove that a generic complex deformation of a generalized Kummer variety contains no complex analytic tori.

Added: Oct 16, 2015

Working paper

This study compares Russian state information systems of industry with UN international recommendations. The harmonization of the Russian information system with successful international practices is necessary for measuring the main industrial indicator levels and dynamics in comparison with the information analogues of both cross-border and strategically important countries. This allows the estimation of the efficiency and competitiveness of Russian industry and the best decisions to be made at all levels of governance in Russia including in the technology and innovation policy sphere. The study shows that a significant number of annual, quarterly, monthly and weekly reporting forms within the Russian statistical system in the absence of a single questionnaire for obtaining comprehensive information from an enterprise and unified methodological recommendations do not solve the information gap problem. The available disaggregated information is not sufficient to analyse the quality and effectiveness of industrial policy, especially in comparison with global levels and tendencies of re-industrialization. The state statistical system needs modernization to reduce the reporting burden on enterprises and improve information transparency and comparability at the detailed level.

Added: Mar 7, 2017

Working paper

On the projective plane there is a unique cubic root of the canonical bundle and this root is acyclic. On fake projective planes such root exists and is unique if there are no 3-torsion divisors (and usually exists but not unique otherwise). Earlier we conjectured that any such cubic root (assuming it exists) must be acyclic. In the present note we give a new short proof of this statement and show acyclicity of some other line bundles on those fake projective planes with at least 9 automorphisms. Similarly to our earlier work we employ simple representation theory for non-abelian finite groups. The novelty stems from the idea that if some line bundle is non-linearizable with respect to a finite abelian group, then it should be linearized by a finite (non-abelian) Heisenberg group. Our argument also exploits J. Rogawski's vanishing theorem and the linearization of an auxiliary line bundle.

Added: Feb 23, 2016

Working paper

For slow–fast quantum systems, we compute first corrections to the quantum action and to the effective slow Hamiltonian.

Added: Apr 9, 2014

Working paper

We show that affine cones over smooth cubic surfaces do not admit non-trivial
$\mathbb{G}_a$ -actions.

Added: Dec 27, 2013

Working paper

A non-degenerate two-dimentional linear operator Ф transforms the unit circle into ellipse.We define the coefficient of deformation k(Ф), as the relation of the lendth of the smaller ellipses axis to its bigger one. In this work we compute the mean value of k(Ф).Analogously, we define the deformation coefficient k(Ф) in three-dimensional case and give an estimation of its mean value.

Added: Jun 28, 2017

Working paper

In this work we demonstrate, how the use of polar decomposition allows one to understand metric proporties of non-degenerate linear operators on the plane. We study how the non-isometric operatpr changes the length of vectors and the angles between the vectors.

Added: Mar 10, 2016

Working paper

The notion of a (metric) modular on an arbitrary set and the corresponding modular space, more general than a metric space, were introduced and studied recently by the author [V.V. Chistyakov, Metric modulars and their application, Dokl. Math. 73 (1) (2006) 32–35, and Modular metric spaces, I: Basic concepts, Nonlinear Anal. 72 (1) (2010) 1–14]. In this paper we establish a fixed point theorem for contractive maps in modular spaces. It is related to contracting rather “generalized average velocities” than metric distances, and the successive approximations of fixed points converge to the fixed points in a weaker sense as compared to the metric convergence.

Added: Feb 6, 2013

Working paper

A comparative analysis of the age impact on happiness in Russia and European countries was conducted. The European Social Survey data in 2012 for 29 countries were used. On the basis of an ordered logistic regression, a U-shape relationship between age and happiness was obtained for some of the analysed countries.
By using cluster analysis, the countries were divided into 3 groups, in which the age effect varies greatly. In the counties of group 1 (for example, Iceland and Norway) happiness did not change at any age or increase smoothly in old age. Group 2 (Germany and France) had a clear U-shaped age-happiness form. Russia and some counties of former Soviet Union: Ukraine, Lithuania and Estonia were analysed in group 3, where the level of happiness decreased significantly in old age (over 60). In some countries (Belgium, Switzerland, Cyprus, Denmark, Finland, Israel, Italy, Sweden) all people were happy, regardless of age and the assumption of age-happiness U-shape relation was not found.
The socio-economic determinants of happiness were also analysed in different age groups. Income satisfaction and subjective health were the more significant characteristics.

Added: Dec 24, 2015

Working paper

Added: Apr 8, 2015

Working paper

In this paper we consider a graphical realization of dynamic programming. The concept is discussed on the partition and knapsack problems. In contrast to dynamic programming, the new algorithm can also treat problems with non-integer data without necessary transformations of the corresponding problem. We compare the proposed method with existing algorithms for these problems on small-size instances of the partition problem with $n \le 10$ numbers. For almost all instances, the new algorithm considers on average substantially less "stages" than the dynamic programming algorithm.

Added: Mar 4, 2013

Working paper

We prove a version of the Aleksandrov-Fenchel inequality for mixed volumes of coconvex bodies. This version is motivated by an inequality from commutative algebra relating intersection multiplicities of ideals.

Added: Oct 6, 2013

Working paper

A projective manifold M is algebraically hyperbolic if there exists a positive constant A such that the degree of any curve of genus g on M is bounded from above by A(g-1). A classical result is that Kobayashi hyperbolicity implies algebraic hyperbolicity. It is known that Kobayashi hyperbolic manifolds have finite automorphism groups. Here we prove that, more generally, algebraically hyperbolic projective manifolds have finite automorphism groups.

Added: Nov 17, 2017

Working paper

We classify all connected affine algebraic groups G such that there are only finitely many G-orbits in every algebraic G-variety containing a dense open G-orbit. We also prove that G enjoys this property if and only if every irreducible algebraic G-variety X is modality-regular, i.e., the modality of X (in the sense of V. Arnol’d) equals to that of a family which is open in X.

Added: Jul 24, 2017

Working paper

We consider single machine problems with opposite criteria, namely we consider the maximization of total tardiness, the maximization of the number of tardy jobs and the maximization of total completion time (in contrast to usual minimization problems)and a minimization version of the Knapsack problem.

Added: Mar 4, 2013

Working paper

The scheduling problem of minimizing total tardiness on a single machine is knownto be NP-hard in the ordinary sense. In this paper, we consider the special case of the problem when the processing times $p_j$ and the due dates $d_j$ of the jobs $j, \, j \in N = \{ 1, 2, \ldots, n \}$, are oppositely ordered: $p_1\ge p_2\ge\dots\ge p_n$ and $d_1\le d_2\le\dots\le d_n$. It is shown that already this special case is $NP$-hard in the ordinary sense, too. The set of jobs $N$ is partitioned into $\Bbbk, 1 \le \Bbbk \le n$, subsets$\mathcal{M}_1,\mathcal{M}_2,\dots,\mathcal{M}_\Bbbk$,$\mathcal{M}_\nu \bigcap \mathcal{M}_\mu=\emptyset$ for $\nu\ne \mu,$$N=\mathcal{M}_1\bigcup\mathcal{M}_2\bigcup\dots\bigcup\mathcal{M}_\Bbbk$,such that$\max_{i,j\in\mathcal{M}_\nu}|d_i-d_j|\le\min_{j\in\mathcal{M}_\nu}p_j$for each $\nu=1,2,\dots,\Bbbk$. We propose algorithms which solve the problem: in $O(\Bbbk n\sum p_j)$ time if $1\le \Bbbk< n$ in $O(n^2)$ time if $\Bbbk= n$ and in $O(n^2)$ time if $\max_{i,j\in N}|d_i-d_j|\le 1$. The polynomial algorithms do neitherrequire the conditions $p_1\ge p_2\ge\dots\ge p_n$ mentioned above nor integer processing times to construct an optimal schedule. Finally, we apply the idea of the presented algorithm for the case $\Bbbk = 1$ to the even-odd partition problem

Added: Mar 4, 2013