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Working paper

This is the first part of a two parts paper dedicated to global bifurcations in the plane. In this part we construct an open set of three parameter families whose topological classification has a numerical invariant that may take an arbitrary positive value. In the second part we construct an open set of six parameter families whose topological classification has a functional invariant. Any germ of a monotonically increasing function may be realized as this invariant. Here "families" are "families of vector fields in the two-sphere".

Added: Jun 24, 2015

Working paper

We study the enumerative geometry of orbits of multidimensional toric action on projective algebraic varieties and develop a new cyclic differential-graded operad, conjecturally governing the real version of the enumerative geometry of these toric orbits.

Added: Nov 18, 2015

Working paper

We prove that a finite 3-group in the Cremona group Cr_3(ℂ) can be generated by at most 4 elements. This provides the last missing piece in bounding the ranks of finite p-subgroups in the space Cremona group.

Added: Feb 10, 2021

Working paper

We prove that for a Q-Gorenstein degeneration $X$ of del Pezzo surfaces, the number of non-Du Val singularities is at most $\rho(X)+2$. Degenerations with $\rho(X)+2$ and $\rho(X)+1$ non-Du Val points are investigated.

Added: Oct 11, 2013

Working paper

We give several characterizations of relative homological epimorphisms in the setting of locally convex topological algebras, thereby correcting a gap in our earlier paper [Trans. Moscow Math. Soc. 2008, 27-104].

Added: Apr 29, 2021

Working paper

We consider an initial-boundary value problem for a 2D time-dependent Schrödinger equation on a semi-infinite strip. For the Numerov-Crank-Nicolson finite-difference scheme with discrete transparent boundary conditions, the Strang-type splitting with respect to the potential is applied. For the resulting method, the uniqueness of a solution and the uniform in time L_2-stability (in particular, L_2-conservativeness) are proved. Due to the splitting, an effective direct algorithm using FFT in the direction perpendicular to the strip is developed to implement the splitting method for general potential. Numerical results on the tunnel effect for smooth and rectangular barriers together with the practical error analysis on refining meshes are included as well.

Added: Jul 24, 2013

Working paper

We provide a nontrivial upper bound for the nonnegative rank of rank-three matrices, which allows us to prove that [6(n+1)/7] linear inequalities suffice to describe a convex n-gon up to a linear projection.

Added: Jun 9, 2013

Working paper

For Fano manifolds we define Ap\'ery constants and Ap\'ery class as particular limits of ratios of coefficients of solutions of the quantum differential equation. We do numerical computations in case of homogeneous varieties. These numbers are identified to be polynomials in the values of Riemann zeta-function with natural arguments.

Added: Apr 19, 2016

Working paper

We show that a direct limit of surjections of (weak) Golod--Shafarevich algebras is a weak Golod--Shafarevich algebra as well. This holds both for graded and for filtered algebras provided that the filtrations are induced by the filtration of the first entry of the sequence. It follows that the limit is an algebra of exponential growth. An example shows that the assumptions of this theorem cannot be directly weakened.

Added: Feb 2, 2015

Working paper

We present an overview of recent results in locally conformally K¨ahler geometry, with focus on the topological properties which obstruct the existence of such structures on compact manifolds.

Added: Nov 2, 2012

Working paper

A "rational" version of the strengthened form of the Commuting Derivation Conjecture, in which the assumption of commutativity is dropped, is proved. A systematic method of constructing in any dimension greater than 3 the examples answering in the negative a question by M. El Kahoui is developed.

Added: Sep 24, 2014

Working paper

We propose a construction of a tensor exact category F_X^m of Artin-Tate motivic sheaves with finite coefficients Z/m over an algebraic variety X (over a field K of characteristic prime to m) in terms of etale sheaves of Z/m-modules over X. Among the objects of F_X^m, in addition to the Tate motives Z/m(j), there are the cohomological relative motives with compact support M_cc^m(Y/X) of varieties Y quasi-finite over X. Exact functors of inverse image with respect to morphisms of algebraic varieties and direct image with compact supports with respect to quasi-finite morphisms of varieties Y\to X act on the exact categories F_X^m. Assuming the existence of triangulated categories of motivic sheaves DM(X,Z/m) over algebraic varities X over K and a weak version of the "six operations" in these categories, we identify F_X^m with the exact subcategory in DM(X,Z/m) consisting of all the iterated extensions of the Tate twists M_cc^m(Y/X)(j) of the motives M_cc^m(Y/X). An isomorphism of the Z/m-modules Ext between the Tate motives Z/m(j) in the exact category F_X^m with the motivic cohomology modules predicted by the Beilinson-Lichtenbaum etale descent conjecture (recently proven by Voevodsky, Rost, et al.) holds for smooth varieties X over K if and only if the similar isomorphism holds for Artin-Tate motives over fields containing K. When K contains a primitive m-root of unity, the latter condition is equivalent to a certain Koszulity hypothesis, as it was shown in our previous paper.

Added: Feb 20, 2014

Working paper

We construct a nontrivial identity which holds in the semigroup of tropical 3-by-3 matrices.

Added: Mar 14, 2015

Working paper

It is well--known that certain
properties of continuous functions on the circle $\mathbb T$,
related to the Fourier expansion, can be improved by a change
of variable, i.e., by a homeomorphism of the circle onto
itself. One of the results in this area is the Jurkat--Waterman
theorem on conjugate functions, which improves the classical
Bohr--Pal theorem. In the present work we provide a short
and technically very simple proof of the Jurkat--Waterman
theorem. Our approach yields a stronger result.

Added: Nov 10, 2016

Working paper

A counterexample to the Jordan-H\"older property for semiorthogonal decompositions of derived categories of smooth projective varieties was constructed by B\"ohning, Graf von Bothmer and Sosna. In this short note we present a simpler example by realizing Bondal's quiver in the derived category of a blowup of the projective space.

Added: Jul 1, 2013

Working paper

Ducomet B.,

, arxiv.org. math. Cornell University, 2013. No. 1309.7280 .
An initial-boundary value problem for the $n$-dimensional ($n\geq 2$) time-dependent Schrödinger equation in a semi-infinite (or infinite) parallelepiped is considered. Starting from the Numerov-Crank-Nicolson finite-difference scheme, we first construct higher order scheme with splitting space averages having much better spectral properties for $n\geq 3$. Next we apply the Strang-type splitting with respect to the potential and, third, construct discrete transparent boundary conditions (TBC). For the resulting method, the uniqueness of solution and the unconditional uniform in time $L^2$-stability (in particular, $L^2$-conservativeness) are proved. Owing to the splitting, an effective direct algorithm using FFT (in the coordinate directions perpendicular to the leading axis of the parallelepiped) is applicable for general potential. Numerical results on the 2D tunnel effect for a P\"{o}schl-Teller-like potential-barrier and a rectangular potential-well are also included.

Added: Oct 1, 2013

Working paper

Motivated by the study of Fano type varieties we define a new class of log pairs that we call asymptotically log Fano varieties and strongly asymptotically log Fano varieties. We study their properties in dimension two under an additional assumption of log smoothness, and give a complete classification of two dimensional strongly asymptotically log smooth log Fano varieties. Based on this classification we formulate an asymptotic logarithmic version of Calabi's conjecture for del Pezzo surfaces for the existence of K\"ahler--Einstein edge metrics in this regime. We make some initial progress towards its proof by demonstrating some existence and non-existence results, among them a generalization of Matsushima's result on the reductivity of the automorphism group of the pair, and results on log canonical thresholds of pairs. One by-product of this study is a new conjectural picture for the small angle regime and limit which reveals a rich structure in the asymptotic regime, of which a folklore conjecture concerning the case of a Fano manifold with an anticanonical divisor is a special case.

Added: Dec 27, 2013

Working paper

We study a topological space obtained from a graph by replacing vertices with smooth Riemannian manifolds, i.e. a decorated graph. We construct a semiclassical asymptotics of the solutions of Cauchy problem for a time-dependent Schroedinger equation on a decorated graph with a localized initial function. The main term of our asymptotic solution at an arbitrary finite time is the sum of Gaussian packets and generalized Gaussian packets. We study the number of such packets as time goes to infinity. We prove asymptotic estimations for this number for the following decorated graphs: cylinder with a segment, two dimensional torus with a segment, three dimensional torus with a segment.

Added: Mar 4, 2014

Working paper

The paper is focused on the existence problem of attractors for foliations. Since the existence of an attractor is a transversal property of the foliation, it is natural to consider foliations admitting transversal geometric structures. As transversal structures are chosen Cartan geometries due to their universality. The existence problem of an attractor on a complete Cartan foliation is reduced to a similar problem for the action of its structure Lie group on a certain smooth manifold. In the case of a complete Cartan foliation with a structure subordinated to a transformation group, the problem is reduced to the level of the global holonomy group of this foliation. Each countable automorphism group preserving a Cartan geometry on a manifold and admitting an attractor is realized as the global holonomy group of some Cartan foliation with an attractor. Conditions on the linear holonomy group of a leaf of a reductive Cartan foliation sufficient for the existence of an attractor (and a global attractor) which is a minimal set are found. Various examples are considered.

Added: Mar 23, 2017

Working paper

For Milnor, statistical, and minimal attractors, we construct examples of smooth flows φ on S^2 for which the attractor of the Cartesian square of φ is smaller than the Cartesian square of the attractor of φ. In the example for the minimal attractors, the flow φ also has an SRB-measure such that its square does not coincide with an SRB-measure of the square of φ.

Added: Nov 12, 2020

Working paper

For a geometrically rational surface X over a perfect field of characteristic different from 2 that contains all roots of 1, we show that either X is birational to a product of a projective line and a conic, or the group of birational automorphisms of X has bounded finite subgroups. As an auxiliary result, we show boundedness of finite subgroups in anisotropic linear algebraic groups of bounded rank and number of connected components. Also, we provide applications to Jordan property for groups of birational automorphisms.

Added: Oct 21, 2018