We calculate characteristic polynomials of operators explicitly represented as polynomials of rank $1$ operators. Applications of the results obtained include a generalization of the Forman--Kenyon's formula for a determinant of the graph Laplacian and also provide its level $2$ analog involving summation over triangulated nodal surfaces with boundary.

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Working paper

We present the generating function for the numbers of isomorphism classes of coverings of the two-dimensional sphere by the genus g compact oriented surface not ramified outside of a given set of m+1 points in the target, fixed ramification type over one point,
and arbitrary ramification types over the remaining m points. We present the genus expansion of this generating function and prove, that the generating function of coverings of genus 0 satisfies some system of differential equations.

Added: Dec 25, 2013

Working paper

For arrangements of n pseudolines in the real projective plane let t_i denote the number of vertices incident to i lines. We obtain a linear on t_i inequality similar to the Hirzebruch one, but with an elementary proof. We construct lower bounds of the number of regions of the complement to the arrangement basing on t_i inequalities. These lower bounds could be used in problems, dealing with numbers of regions in the complement, as N. Martinov had used.

Added: Mar 25, 2014

Working paper

We consider the systems of diffusion-orthogonal polynomials, defined in the
work [1] of D. Bakry, S. Orevkov and M. Zani and (particularly) explain why
these systems with boundary of maximal possible degree should always come from
the group, generated by reflections. Our proof works for the dimensions $2$ (on
which this phenomena was discovered) and $3$, and fails in the dimensions $4$
and higher, leaving the possibility of existence of diffusion-orthogonal
systems related to the Einstein metrics.
The methods of our proof are algebraic / complex analytic in nature and based
mainly on the consideration of the double covering of $\mathbb{C}^d$, branched
in the boundary divisor.
Author wants to thank Stepan Orevkov, Misha Verbitsky and Dmitry Korb for
useful discussions.

Added: Sep 19, 2014

Working paper

We consider the Cauchy problem for the 1D generalized Schrödinger equation on the whole axis. To solve it, any order finite element in space and the Crank-Nicolson in time method with the discrete transpa\-rent boundary conditions (TBCs) has recently been constructed. Now we engage the Richardson extrapolation to improve significantly the accuracy in time step. To study its properties, we give results of numerical experiments and enlarged practical error analysis for three typical examples. The resulting method is able to provide high precision results in the uniform norm for reasonable computational costs that is unreachable by more common 2nd order methods in either space or time step. Comparing our results to the previous ones, we obtain much more accurate results using much less amount of both elements and time steps.

Added: May 14, 2014

Working paper

Bossy M.,

, Martinez K. arxiv.org. math. Cornell University, 2020
We consider the problem of the approximation of the solution of a one-dimensional SDE with non-globally Lipschitz drift and diffusion coefficients behaving as xα, with α>1. We propose an (semi-explicit) exponential-Euler scheme and study its convergence through its weak approximation error. To this aim, we analyze the C1,4 regularity of the solution of the associated backward Kolmogorov PDE using its Feynman-Kac representation and the flow derivative of the involved processes. From this, under some suitable hypotheses on the parameters of the model ensuring the control of its positive moments, we recover a rate of weak convergence of order one for the proposed exponential Euler scheme. Finally, numerical experiments are shown in order to support and complement our theoretical result.

Added: Sep 24, 2020

Working paper

We investigate basic properties of uniformly rational varieties, i.e. those smooth varieties for which every point has a Zariski open neighborhood isomorphic to an open subset of A^n. It is an open question of Gromov whether all smooth rational varieties are uniformly rational. We discuss some potential criteria that might allow one to show that they form a proper subclass in the class of all smooth rational varieties. Finally we prove that small algebraic resolutions and big resolutions of nodal cubic threefolds are uniformly rational.

Added: Oct 9, 2013

Working paper

We characterize open embeddings of Stein spaces and of $C^\infty$-manifolds in terms of certain flatness-type conditions on the respective homomorphisms of function algebras.

Added: Aug 7, 2019

Working paper

The notions of consistent pairs and consistent chains of t-structures are introduced. A theorem that two consistent chains of t-structures generate a distributive lattice is proven. The technique developed is then applied to the pairs of chains obtained from the standard t-structure on the derived category of coherent sheaves and the dual t-structure by means of the shift functor. This yields a family of t-structures whose hearts are known as perverse coherent sheaves.

Added: Dec 27, 2013

Working paper

Added: May 15, 2012

Working paper

We consider a control problem for longitudinal vibrations of a nonhomogeneous bar with clamped ends. The vibrations of the bar are controlled by an external force which is distributed along the bar. For the minimization problem of mean square deviation of the bar we prove that the optimal control has an infinite number of switchings in a finite time interval, i.e., the optimal control is the chattering control.

Added: Feb 20, 2013

Working paper

In this paper, an SIR epidemic model with variable size of population is considered. We study optimal control problem for an SIR model with "vaccination" and "treatment" as controls. It is shown that an optimal control exists. We have already used functional, that lots of researchers use, and found that this functional is not appropriate, it has a defect. Now, we xed this defect by changing this functional. We analyze the dependence of solutions on parameter of problems and discuss our result.

Added: Dec 15, 2019

Working paper

We consider probability measures on $\mathbb{R}^{\infty}$ and study natural analogs of optimal transportation mappings for the case of infinite Kantorovich distance. Our examples include 1) quasi-product measures, 2) measures with certain symmetric properties, in particular, exchangeable and stationary measures. It turns out that the existence problem for optimal transportation is closely related to various ergodic properties. We prove the existence of optimal transportation for a certain class of stationary Gibbs measures. In addition, we establish a variant of the Kantorovich duality for the Monge--Kantorovich problem restricted to the case of measures invariant with respect of actions of compact groups.

Added: May 13, 2013

Working paper

We study the properties of a sequence c_n defined by the recursive relation
c_0/(n+1)+c_1/(n+2)+…+c_n/(2n+1)=0
for n>1 and c_0=1. This sequence also has an alternative definition in terms of certain norm minimization in the space L^2([0,1]). We prove estimates on growth order of c_n and the sequence of its partial sums, infinite series identities, connecting c_n with harmonic numbers H_n and also formulate some conjectures based on numerical computations.

Added: Jul 17, 2019

Working paper

Gamayun, Iorgov and Lisovyy in 2012 proposed that tau function of the Painlevé equation equals to the series of c=1 Virasoro conformal blocks. We study similar series of c=−2 conformal blocks and relate it to Painlevé theory. The arguments are based on Nakajima-Yoshioka blow-up relations on Nekrasov partition functions.
We also study series of q-deformed c=−2 conformal blocks and relate it to q-Painlevé equation. Using this we prove formula for the tau function of q-Painlevé A_7^{(1)′} equation.

Added: Nov 22, 2018

Working paper

Aminov S., Arthamonov S.,

et al. arxiv.org. math. Cornell University, 2013
We propose multidimensional versions of the Painleve VI equation and its degenerations. These field theories are related to the isomonodromy problems on flat holomorphic infinite rank bundles over elliptic curves and take the form of non-autonomous Hamiltonian equations. The modular parameter of curves plays the role of "time". Reduction of the field equations to the zero modes leads to SL(N,C) monodromy preserving equations. In particular, the latter coincide with the Painleve VI equation for N=2. We consider two types of the bundles. In the first one the group of automorphisms is the centrally and cocentrally extended loop group L(SL(N,C)) or some multiloop group. In the case of the Painleve VI field theory in D=1+1 the four constants of the Painleve VI equation become dynamical fields. The bundles of the second type are defined by the group of automorphisms of the noncommutative torus. They lead to the equations in dimension 2+1. In both cases we consider trigonometric, rational and scaling limits of the theories. Generically (except some degenerate cases) the derived equations are nonlocal. We consider Whitham quasiclassical limit to integrable systems. In this way we derive two and three dimensional integrable nonlocal versions of the integrable Euler-Arnold tops.

Added: Dec 27, 2013

Working paper

We describe all reductive spherical subgroups of the group SL(n) which have connected intersection with any parabolic subgroup of the group SL(n). This condition guarantees that any open equivariant embedding of the corresponding homogeneous space into a Moishezon space is algebraic.

Added: Oct 14, 2013

Working paper

It is a classical result of Euler that the rotation of a torque-free three-dimensional rigid body about the short or the long axis is stable, and the rotation about the middle axis is unstable. This result is generalized to the case of a multidimensional body.

Added: Nov 19, 2013

Working paper

We construct an example of a one-dimensional parabolic integro-differential equation with nonlocal diffusion which does not have asymptotically finite-dimensional dynamics in the corresponding state space. This example is more natural in the class of evolutionary equations of parabolic type than those known earlier.

Added: Nov 18, 2013

Working paper

The classical Peter-Weyl theorem describes the structure of the space of functions on a semi-simple algebraic group. On the level of characters (in type A) this boils down to the Cauchy identity for the products of Schur polynomials. We formulate and prove the analogue of the Peter-Weyl theorem for the current groups. In particular, in type A the corresponding characters identity is governed by the Cauchy identity for the products of q-Whittaker functions. We also formulate and prove a version of the Schur-Weyl theorem for current groups. The link between the Peter-Weyl and Schur-Weyl theorems is provided by the (current version of) Howe duality.

Added: Jun 20, 2019

Working paper

Schubert polynomials for the classical groups were defined by S.Billey and M.Haiman in 1995; they are polynomial representatives of Schubert classes in a full flag variety of a classical group. We provide a combinatorial description for these polynomials, as well as their double versions, by introducing analogues of pipe dreams, or RC-graphs, for the Weyl groups of the classical types.

Added: Sep 30, 2020

Working paper

We construct special rational ${\rm gl}_N$ Knizhnik-Zamolodchikov-Bernard
(KZB) equations with $\tilde N$ punctures by deformation of the corresponding
quantum ${\rm gl}_N$ rational $R$-matrix. They have two parameters. The limit
of the first one brings the model to the ordinary rational KZ equation. Another
one is $\tau$. At the level of classical mechanics the deformation parameter
$\tau$ allows to extend the previously obtained modified Gaudin models to the
modified Schlesinger systems. Next, we notice that the identities underlying
generic (elliptic) KZB equations follow from some additional relations for the
properly normalized $R$-matrices. The relations are noncommutative analogues of
identities for (scalar) elliptic functions. The simplest one is the unitarity
condition. The quadratic (in $R$ matrices) relations are generated by
noncommutative Fay identities. In particular, one can derive the quantum
Yang-Baxter equations from the Fay identities. The cubic relations provide
identities for the KZB equations as well as quadratic relations for the
classical $r$-matrices which can be halves of the classical Yang-Baxter
equation. At last we discuss the $R$-matrix valued linear problems which
provide ${\rm gl}_{\tilde N}$ Calogero-Moser (CM) models and Painleve equations
via the above mentioned identities. The role of the spectral parameter plays
the Planck constant of the quantum $R$-matrix. When the quantum ${\rm gl}_N$
$R$-matrix is scalar ($N=1$) the linear problem reproduces the Krichever's
ansatz for the Lax matrices with spectral parameter for the ${\rm gl}_{\tilde
N}$ CM models. The linear problems for the quantum CM models generalize the KZ
equations in the same way as the Lax pairs with spectral parameter generalize
those without it.

Added: Jan 23, 2015