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Working paper

Let $\gamma$ be a Gaussian measure on a locally convex space and $H$ be the corresponding Cameron-Martin space. It has been recently shown by L. Ambrosio and A. Figalli that the linear first-order PDE $$ \dot{\rho} + \mbox{div}_{\gamma} (\rho \cdot {b}) =0, \ \ \rho|_{t=0} = \rho_0, $$ where $\rho_0 \cdot \gamma $ is a probability measure, admits a weak solution, in particular, under the following assumptions: $$ \|b\|_{H} \in L^p(\gamma), \ p>1, \ \ \ \exp\bigl(\varepsilon(\mbox{\rm div}_{\gamma} b)_{-} \bigr) \in L^1(\gamma). $$ Applying transportation of measures via triangular maps we prove a similar result for a large class of non-Gaussian probability measures $\nu$ on $\R^{\infty}$, under the main assumption that $\beta_i \in \cap_{n \in \Nat} L^{n}(\nu)$ for every $i \in \Nat$, where $\beta_i$ is the logarithmic derivative of $\nu$ along the coordinate $x_i$. We also show uniqueness of the solution for a wide class of measures. This class includes uniformly log-concave Gibbs measures and certain product measures.

Added: May 13, 2013

Working paper

Characteristic curves of a Hamilton–Jacobi equation can be seen as action minimizing trajectories of fluid particles. However this description is valid only for smooth solutions. For nonsmooth “viscosity” solutions, which give rise to discontinuous velocity fields, this picture holds only up to the moment when trajectories hit a shock and cease to minimize the Lagrangian action. In this paper we show that for any convex Hamiltonian, a viscous regularization allows to construct a nonsmooth flow that extends particle tra- jectories and determines dynamics inside the shock manifolds. This flow consists of integral curves of a particular velocity field, which is uniquely defined everywhere in the flow domain and is discontinuous on shock manifolds.

Added: Feb 5, 2013

Working paper

We consider planar cubic maps, i.e. connected cubic graphs, embedded into plane, with marked spanning tree and marked directed edge (not in this tree). The number of such objects with 2n vertices is C2n∙ Cn+1, where Ck is Catalan number.

Added: Sep 1, 2016

Working paper

Consider a finite group $G$ acting on a triangulated category $\TTT$. In this paper we try to understand when the category $\TTT^G$ of $G$-equivariant objects in $\TTT$ is triangulated. We prove that it is so in two cases: the action on the derived category $\D^b(\AA)$ induced by an action on an abelian category $\AA$ and the action on the homotopy category $H^0(\AA)$ of a pretriangulated DG-category $\AA$ induced by an action on $\AA$. Also, we show that the relation ``to be an equivariant category with respect to a finite abelian group action'' is symmetric on idempotent complete additive categories.

Added: Sep 15, 2014

Working paper

We deal with an initial-boundary value problem for the generalized time-dependent Schr\"odinger equation with variable coefficients in an unbounded $n$--dimensional parallelepiped ($n\geq 1$). To solve it, the Crank-Nicolson in time and the polylinear finite element in space method with the discrete transpa\-rent boundary conditions is considered. We present its stability properties and derive new error estimates $O(\tau^2+|h|^2)$ uniformly in time in $L^2$ space norm, for $n\geq 1$, and mesh $H^1$ space norm, for $1\leq n\leq 3$ (a superconvergence result), under the Sobolev-type assumptions on the initial function. Such estimates are proved for methods with the discrete TBCs for the first time.

Added: Mar 27, 2015

Working paper

Given a connected reductive complex algebraic group G and a spherical subgroup H⊂G, the extended weight monoid Λˆ+G(G/H) encodes the G-module structures on spaces of regular sections of all G-linearized line bundles on G/H. Assuming that G is semisimple and simply connected and H is specified by a regular embedding in a parabolic subgroup P⊂G, in this paper we obtain a description of Λˆ+G(G/H) via the set of simple spherical roots of~G/H together with certain combinatorial data explicitly computed from the pair (P,H). As an application, we deduce a new proof of a result of Avdeev and Gorfinkel describing Λˆ+G(G/H) in the case where H is strongly solvable.

Added: Jun 9, 2020

Working paper

Added: Oct 15, 2015

Working paper

We introduce a family of linear relations between cell-zeta values that have a form similar to product map relations and jointly with them imply stuffle relations between multiple zeta values.

Added: Oct 16, 2020

Working paper

Given an operad P with a finite Grobner basis of relations, we study the generating functions for the dimensions of its graded components P(n). Under moderate assumptions on the relations we prove that the exponential generating function for the sequence {dim P(n)} is differential algebraic, and in fact algebraic for P is a symmetrization of a non-symmetric operad. If, in addition, the growth of the dimensions P(n) is bounded by an exponent of n (or a polynomial of n, in the non-symmetric case) then, Moreover, the ordinary generating function for the above sequence {dim P(n)} is rational. We give a number of examples of calculations and discuss conjectures about the above generating functions for more general classes of operads.

Added: May 13, 2014

Working paper

For a point p in a complex projective plane and a triple (g,d,l) of non-negative
integers we define a plane Hurwitz number of the Severi variety
W_{g,d,l} consisting of all reduced irreducible plane curves of
genus g and degree d+l having an l-fold node at p and at
most ordinary nodes as singularities at the other points. In the cases
d+l >= g+2 and d+2l >= g+2 > d+l we express the plane
Hurwitz numbers via appropriate ordinary Hurwitz numbers. The
remaining case d+2l<g+2 is still widely open.

Added: Jul 5, 2016

Working paper

Added: Nov 13, 2017

Working paper

We explore orbits, rational invariant functions, and quotients of the natural actions of connected, not necessarily finite dimensional subgroups of the automorphism groups of irreducible algebraic varieties. The applications of the results obtained are given.

Added: Jan 3, 2014

Working paper

We describe the geometry of K\"uchle varieties (i.e. Fano 4-folds of index 1 contained in the Grassmannians as zero loci of equivariant vector bundles) with Picard number greater than 1 and the structure of their derived categories.

Added: Jan 30, 2015

Working paper

We consider bounded analytic functions in domains generated by sets that have Littlewood--Paley property. We show that each such function is an lp -multiplier.

Added: Feb 16, 2014

Working paper

We study cubic rational maps that take lines to plane curves. A complete description of such cubic rational maps concludes the classification of all planarizations, i.e., maps taking lines to plane curves.

Added: Sep 16, 2014

Working paper

We construct two Lefschetz decompositions of the derived category of coherent sheaves on the Grassmannian of $k$-dimensional subspaces in a vector space of dimension $n$. Both of them admit a Lefschetz basis consisting of equivariant vector bundles. We prove fullness of the first decomposition and conjecture it for the second one. In the case when $n$ and $k$ are coprime these decompositions coincide and are minimal. In general, we conjecture minimality of the second decomposition.

Added: Oct 10, 2013

Working paper

We consider the class of singular double coverings $X \to \PP^3$ ramified in the degeneration locus $D$ of a family of 2-dimensional quadrics. These are precisely the quartic double solids constructed by Artin and Mumford as examples of unirational but nonrational conic bundles. With such quartic surface $D$ one can associate an Enriques surface $S$ which is the factor of the blowup of $D$ by a natural involution acting without fixed points (such Enriques surfaces are known as nodal Enriques surfaces or Reye congruences). We show that the nontrivial part of the derived category of coherent sheaves on this Enriques surface $S$ is equivalent to the nontrivial part of the derived category of a minimal resolution of singularities of $X$.

Added: Oct 4, 2013

Working paper

In this work, we obtain some new lower bounds for the number N_B(x) of Novak numbers less than or equal to x. We also prove, conditionally on Generalized Riemann Hypothesis, the upper estimates for the number of primes dividing at least one Nov\'ak number and give description for the prime factors of Nov\'ak numbers N such that 2N is a Novak-Carmichael number.

Added: Oct 19, 2017

Working paper

Kharlamov V.,

. arxiv.org. math. Cornell University, 2013
In this article, we investigate some properties of cyclic coverings of complex surfaces of general type branched along smooth curves that are numerically equivalent to a multiple of the canonical class. The main results concern coverings of surfaces of general type with p_g=0 and Miyaoka--Yau surfaces; in particular, they provide new examples of multicomponent moduli spaces of surfaces with given Chern numbers as well as new examples of surfaces that are not deformation equivalent to their complex conjugates.

Added: Dec 27, 2013

Working paper

The famous conjecture of V.Ya.Ivrii (1978) says that in every billiard with infinitely-smooth boundary in a Euclidean space the set of periodic orbits has measure zero. In the present paper we study the complex version of Ivrii's conjecture for odd-periodic orbits in planar billiards, with reflections from complex analytic curves. We prove positive answer in the following cases: 1) triangular orbits; 2) odd-periodic orbits in the case, when the mirrors are algebraic curves avoiding two special points at infinity, the so-called isotropic points. We provide immediate applications to the real piecewise-algebraic Ivrii's conjecture and to its analogue in the invisibility theory.

Added: Sep 29, 2013

Working paper

The paper suggests a new --- to the best of the author's knowledge --- characterization of decisions which are optimal in the multi-objective optimization problem with respect to a definite proper preference cone, a Euclidean cone with a prescribed angular radius. The main idea is to use the angle distances between the unit vector and points of utility space.A necessary and sufficient condition for the optimality in the form of an equation is derived. The first-order necessary optimality conditions are also obtained.

Added: Jan 11, 2014