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Working paper

In this paper we give a complete topological classification of orientation preserving Morse-Smale diffeomorphisms on orientable closed surfaces. For MS diffeomorphisms with relatively simple behaviour it was known that such a classification
can be given through a directed graph, a three-colour directed graph or by a certain topological object, called a scheme. Here we will assign to {any} MS surface diffeomorphism a finite amount of data
which completely determines its topological conjugacy class. Moreover, we show that associated to any abstract version of this data, there exists a unique conjugacy class of MS orientation preserving diffeomorphisms (on some orientation preserving surface).
As a corollary we obtain a different proof that nearby MS diffeomorphisms are topologically conjugate.

Added: Dec 7, 2017

Working paper

We present here continued fraction for Zeta(3) parametrized by some family of points (F,G) on projective line. This family of points can be obtained if from full projective line would be removed some no more than countable nowhereмножество dense exeptional set of finite points. countable nowhere dense set, which contains the above exeptional set of finite points, is specified also.

Added: Jan 8, 2014

Working paper

Our main object of study is a 3−valent graph with a vector function on its edges. The function assignes to an edge a pair of 2−adic integer numbers and satisfies additional condition: the sum of its values on three edges, terminating in the same vertex, is equal to 0. For each vertex of the graph three vectors corresponding to these edges generate a lattice over the ring of 2−adic integers. In this paper we study the restrictions, imposed on these lattices by the combinatorics of the graph. As an application we obtain the following fact: a rational balanced polygon cannot be cut into an odd number of triangles of equal areas. First result of this type was obtained by Paul Monsky in 1970. He proved that a square cannot be cut into an odd number of triangles of equal areas. In 2000 Sherman Stein conjectured that the same holds for any balanced polygon. We prove this conjecture in the case, when coordinates of all vertices of the cut are rational numbers.

Added: Nov 15, 2014

Working paper

We construct an analog of the subalgebra $U\mathfrak{gl}(n)\otimes U\mathfrak{gl}(m)\subset U\mathfrak{gl}(m+n)$ in the setting of quantum toroidal algebras and study the restrictions of various representations to this subalgebra.

Added: Apr 24, 2014

Working paper

A polynomial with exactly two critical values is called a generalized Chebyshev polynomial. A polynomial with exactly three critical values is called a Zolotarev polynomial. Two Chebyshev polynomials $f$ and $g$ are called Z-homotopic, if there exists a family $p_\alpha$, $\alpha\in [0,1]$, where $p_0=f$, $p_1=g$ and $p_\alpha$ is a Zolotarev polynomial, if $\alpha\in (0,1)$. As each Chebyshev polynomial defines a plane tree (and vice versa), Z-homotopy can be defined for plane trees. In this work we prove some necessary geometric conditions for plane trees Z-homotopy, describe Z-homotopy for trees with 5 and 6 edges and study one interesting example in the class of trees with 7 edges.

Added: Feb 24, 2013

Working paper

Added: Oct 20, 2017

Working paper

We considered the control problem for wireless sensor networks with a single time server node and a large number of client nodes. The cost functional of this control problem accumulates clock synchronization errors in the clients nodes and the energy consumption of the server over some time interval $[0,T]$. For all possible parameter values we found the structure of optimal control function. It was proved that for any optimal solution $\widehat{R}\left(t\right)$ there exist a time moment $\tau,$ $0\leq\tau<T$, such that $\hat{u}(t)=0,$ $t\in[\tau,T]$, i.e., the sending messages at times close to $T$ is not optimal. We showed that for sufficiently large $u_{1}$ the optimal solutions contain singular arcs. We found conditions on the model parameters under which different types of the optimal control are realized.

Added: Aug 28, 2014

Working paper

By weighted tree we understand such connected tree, that: a) each its vertex and each edge have a positive integer weight; b) the weight of each vertex is equal to the sum of weights of outgoing edges. Each tree has a binary structure --- we can color its vertices in two colors, black and white so, that adjacent vertices have different colors. A type is a set of pairwise non-isotopic plane weighted trees with a given list of weights of white vertices and a given list of weights of black vertices. In this work we present a method for computing the cardinality of a given type.

Added: Oct 29, 2013

Working paper

Agents vote to choose a fair mixture of public outcomes; each agent likes or dislikes each outcome. We discuss three outstanding voting rules. The Conditional Utilitarian rule, a variant of the random dictator, is Strategyproof and guarantees to any group of like-minded agents an influence proportional to its size. It is easier to compute and more efficient than the familiar Random Priority rule. Its worst case (resp. average) inefficiency is provably (resp. in numerical experiments) low if the number of agents is low. The efficient Egalitarian rule protects similarly individual agents but not coalitions. It is Excludable Strategyproof: I do not want to lie if I cannot consume outcomes I claim to dislike. The efficient Nash Max Product rule offers the strongest welfare guarantees to coalitions, who can force any outcome with a probability proportional to their size. But it fails even the excludable form of Strategyproofness.

Added: Oct 31, 2018

Working paper

We will consider a totally real Galois field $K$ of degree 4 as the linear coordinate space $\mathbb{Q}^4\subset\mathbb{R}^4$.
An element $k\in K$ is called strictly positive, if all its conjugates are positive. The set of strictly positive elements is a convex cone in $K$. The convex hull of strictly positive integral elements is a convex subset of this cone and its boundary $\Gamma$ is an infinite union of 3-dimensional polyhedrons. The group $U$ of strictly positive units acts on $\Gamma$: the action of a strictly positive unit permutes polyhedrons. Fundamental domains of this action are the object of study in this work. We mainly present some interesting examples.

Added: Jun 28, 2013

Working paper

Vehicle Routing Problem is a well-known problem in logistics and trans- portation, and the variety of such problems is explained by the fact that it occurs in many real-life situations. It is an NP-hard combinatorial optimization problem and finding an exact optimal solution is practically impossible. In this work, Site- Dependent Truck and Trailer Routing Problem with hard and soft Time Windows and Split Deliveries is considered (SDTTRPTWSD). In this article, we develop a heuristic with the elements of Tabu Search for solving SDTTRPTWSD. The heuris- tic uses the concept of neighborhoods and visits infeasible solutions during the search. A greedy heuristic is applied to construct an initial solution.

Added: Oct 17, 2016

Working paper

We consider the space $\mathcal{M}_{2,1}$ --- the open moduli space of complex curves of genus 2 with one marked point. Using language of chord diagrams we describe the cell structure of $\mathcal{M}_{2,1}$ and cell adjacency. This allows one to construct matrices of boundary operators and compute Betty numbers of $\mathcal{M}_{2,1}$ over $\mathbb{Q}$.

Added: Feb 24, 2013

Working paper

Let M be a compact hyperk¨ahler manifold with maximal holonomy (IHS). The group H2 (M, R) is equipped with a quadratic form of signature (3, b2 − 3), called BogomolovBeauville-Fujiki (BBF) form. This form restricted to the rational Hodge lattice H1,1 (M, Q), has signature (1, k). This gives a hyperbolic Riemannian metric on the projectivisation of the positive cone in H1,1 (M, Q), denoted by H. Torelli theorem implies that the Hodge monodromy group Γ acts on H with finite covolume, giving a hyperbolic orbifold X = H/Γ. We show that there are finitely many geodesic hypersurfaces which cut X into finitely many polyhedral pieces in such a way that each of these pieces is isometric to a quotient P(M′ )/ Aut(M′ ), where P(M′ ) is the projectivization of the ample cone of a birational model M′ of M, and Aut(M′ ) the group of its holomorphic automorphisms. This is used to prove the existence of nef isotropic line bundles on a hyperk¨ahler birational model of a simple hyperk¨ahler manifold of Picard number at least 5, and also illustrates the fact that an IHS manifold has only finite

Added: Nov 10, 2015

Working paper

In this paper, we tackle a problem of predicting phenotypes from structural connectomes. We propose that normalized Laplacian spectra can capture structural properties of brain networks, and hence graph spectral distributions are useful for a task of connectome-based classification. We introduce a kernel that is based on earth mover's distance (EMD) between spectral distributions of brain networks. We access performance of an SVM classifier with the proposed kernel for a task of classification of autism spectrum disorder versus typical development based on a publicly available dataset. Classification quality (area under the ROC-curve) obtained with the EMD-based kernel on spectral distributions is 0.71, which is higher than that based on simpler graph embedding methods.

Added: Dec 9, 2016

Working paper

We study bipartite maps on the plane with one infinite face and one face of perimeter 2. At first we consider the problem of their enumeration and then the connection between the combinatorial structure of a map and the degree of its definition field. The second problem is considered when the number of edges is p+1, where p is a prime.

Added: Jul 5, 2017

Working paper

A numerical solution of differential equation is adequate if, in particular, the obtained discrete model is topologically conjugate to the time one map of the original flow. B.~Garay showed that the Runge-Kutta discretization of a gradient-like flow $(n > 2)$ on the $n$-disk is topologically conjugate to the time one map for a sufficiently small step size. J.~Palis found necessary conditions for a Morse-Smale diffeomorphism on a closed $n$-dimensional manifold $M^n$ to embed into a topological flow and proved that these conditions are also sufficient for $n=2$. We find sufficient conditions for a Morse-Smale diffeomorphism to embed in a topological flow for the case when $M^n$ is the sphere $S^n, \,n\geq 4$.

Added: Oct 13, 2018

Working paper

The question under consideration is Gevrey summability of power expansions of solutions to the third and fifth Painlev\'{e} equations near infinity. Methods of French and Japaneese schools are used to analyse these properties of formal power series solutions. The results obtained are compared with the ones obtained by means of Power Geometry.

Added: Oct 20, 2013

Working paper

A type is the set of all pairwise nonisotopic plane binary trees with the same passport. A type is called decomposable, if it is a union of several Galois orbits. In this work we present the list of all passports of plane binary trees with ten edges and the list of all Galois orbits.

Added: Dec 19, 2014

Working paper

In this article we use the modular decomposition technique for exact solving the weighted maximum clique problem. Our algorithm takes the modular decomposition tree from the paper of Tedder et. al. and finds solution recursively. Also, we propose algorithms to construct graphs with modules. We show some interesting results, comparing our solution with Ostergard's algorithm on DIMACS benchmarks and on generated graphs

Added: Oct 15, 2017