Add a filter

Of all publications in the section: 482

Sort:

by name

by year

Working paper

Abstract. The method of Chernoff approximation was discovered by Paul Chernoff in 1968 and now is a powerful and flexible tool of contemporary functional analysis. This method is different from grid-based approach and helps to solve numerically the Cauchy problem for evolution equations, e.g., for heat equation and for more general parabolic second-order partial differential equations with variable coefficients. The case of heat equation is well-studied and does not require any approximations because Green's function is known in that case; but for more general equations exact formulas for solutions are unknown, so numerical approximations of solutions are often requested by engineers and other researchers dealing with PDEs. Chernoff approximations are functions defined by explicit expressions that contain variable coefficients of the equation and initial condition as parameters. Traditionally researchers construct Chernoff approximations with the use of integral operators which lead to Feynman formulas that are connected with representation of solution in terms of Feynman path integral and Feynman-Kac formulas. This traditional approach has two features that restrict its practical use. First, one needs to integrate over the Cartesian product of $n$ copies of the real line hence $n$-tuple integrals are improper (for better quality of approximation one needs to take larger values of $n$, and this leads to improper integrals of high multiplicity which are difficult to handle in practice). Second, the speed of convergence usually is not very high (not better than $1/n$).
In the present paper we construct Chernoff approximations of a new kind which are free from these disadvantages: all integrals in our formulas are over the segment $[-1,1]$, and the speed of convergence is higher than $1/n$ for initial conditions that are smooth enough. Our approximations provide solution in a form of quasi-Feynman formula which is not clearly connected with Feynman integral but is much better for practical purposes.

Added: Dec 14, 2020

Working paper

We give a combinatorial description of the rational cohomology of the moduli spaces of pointed genus 1 curves with $n$ marked points and level $N$ structures. More precisely, we explicitly describe the $E_2$ term of the Leray spectral sequence of the forgetful mapping $\EuScript{M}_{1,n}(N)\to\EuScript{M}_{1,1}(N)$ and show that the result is isomorphic to the rational cohomology of $\EuScript{M}_{1,n}(N)$ as a rational mixed Hodge structure equipped with an action of the symmetric group $\mathfrak{S}_n$. The classical moduli space $\EuScript{M}_{1,n}$ is the particular case N=1

Added: Mar 25, 2013

Working paper

Added: Feb 6, 2013

Working paper

Let $\bbk$ be a field of characteristic zero and $G$ be a finite group of automorphisms of projective plane over $\bbk$. Castelnuovo's criterion implies that the quotient of projective plane by $G$ is rational if the field $\bbk$ is algebraically closed. In this paper we prove that $\mathbb{P}^2_{\bbk} / G$ is rational for an arbitrary field $\bbk$ of characteristic zero.

Added: Oct 31, 2013

Working paper

A real polynomial p of degree n is called a Morse polynomial if its derivative has n−1 pairwise differentreal roots and values of p in these roots (critical values) are also pairwise different. The plot of such polynomial is called a "snake". By enumerating critical points and critical values in the increasing order we construct a permutation a1,…,an−1, where ai is the number of polynomial's value in i-th critical point. This permutation is called the \emph{passport} of the snake (polynomial). In this work for Morse polynomials of degrees 5 and 6 we describe the partition of the coefficient space into domains of constant passport.

Added: Nov 8, 2019

Working paper

We consider the projective arc schemes of the Veronese embeddings of the flag
varieties for simple Lie groups of type ADE. The arc schemes are not reduced
and we consider the homogeneous coordinate rings of the corresponding reduced
schemes. We show that each graded component of a homogeneous coordinate ring is
a cocyclic module of the current algebra and is acted upon by the algebra of
symmetric polynomials. We show that the action of the polynomial algebra is
free and that the localization at the special point of a graded component is
isomorphic to an affine Demazure module whose level is the degree of the
Veronese embedding. In type $A_1$ we give the precise list of generators of the
reduced arc scheme structure of the Veronese curves. In general type we
introduce the notion of the global higher level Demazure modules and identify
the graded components of the homogeneous coordinate rings with these modules.

Added: Dec 18, 2019

Working paper

Categorical resolution of singularities has been constructed in arXiv:1212.6170. It proceeds by alternating two steps of seemingly different nature. We show how to use the formalism of filtered derived categories to combine the two steps into one. This results in a certain rather natural categorical refinement of the usual blowup of an algebraic variety in a closed subscheme.

Added: Dec 29, 2014

Working paper

We prove that the existence of a strongly reflective modular form of a large weight implies that the Kodaira dimension of the corresponding modular variety is negative or, in some special case, it is equal to zero. Using the Jacobi lifting we construct three towers of strongly reflective modular forms with the simplest possible divisor. In particular we obtain a Jacobi lifting construction of the Borcherds-Enriques modular form Phi_4 and Jacobi liftings of automorphic discriminants of the K\"ahler moduli of Del Pezzo surfaces constructed recently by Yoshikawa. We obtain also three modular varieties of dimension 4, 6 and 7 of Kodaira dimension 0.

Added: Mar 3, 2015

Working paper

Let $\mu = e^{-V} \ dx$ be a probability measure and $T = \nabla \Phi$ be the optimal transportation mapping pushing forward $\mu$ onto a log-concave compactly supported measure $\nu = e^{-W} \ dx$. In this paper, we introduce a new approach to the regularity problem for the corresponding Monge--Amp{\`e}re equation $e^{-V} = \det D^2 \Phi \cdot e^{-W(\nabla \Phi)}$ in the Besov spaces $W^{\gamma,1}_{loc}$. We prove that $D^2 \Phi \in W^{\gamma,1}_{loc}$ provided $e^{-V}$ belongs to a proper Besov class and $W$ is convex. In particular, $D^2 \Phi \in L^p_{loc}$ for some $p>1$. Our proof does not rely on the previously known regularity results.

Added: Mar 28, 2013

Working paper

e describe classical top-like integrable systems arising from the quantum
exchange relations and corresponding Sklyanin algebras. The Lax operator is
expressed in terms of the quantum non-dynamical $R$-matrix even at the
classical level, where the Planck constant plays the role of the relativistic
deformation parameter in the sense of Ruijsenaars and Schneider (RS). The
integrable systems (relativistic tops) are described as multidimensional Euler
tops, and the inertia tensors are written in terms of the quantum and classical
$R$-matrices. A particular case of ${\rm gl}_N$ system is gauge equivalent to
the $N$-particle RS model while a generic top is related to the spin
generalization of the RS model. The simple relation between quantum
$R$-matrices and classical Lax operators is exploited in two ways. In the
elliptic case we use the Belavin's quantum $R$-matrix to describe the
relativistic classical tops. Also by the passage to the noncommutative torus we
study the large $N$ limit corresponding to the relativistic version of the
nonlocal 2d elliptic hydrodynamics. Conversely, in the rational case we obtain
a new ${\rm gl}_N$ quantum rational non-dynamical $R$-matrix via the
relativistic top, which we get in a different way -- using the factorized form
of the RS Lax operator and the classical Symplectic Hecke (gauge)
transformation. In particular case of ${\rm gl}_2$ the quantum rational
$R$-matrix is 11-vertex. It was previously found by Cherednik. At last, we
describe the integrable spin chains and Gaudin models related to the obtained
$R$-matrix.

Added: Jan 23, 2015

Working paper

Klartag B.,

arxiv.org. math. Cornell University, 2016
According to a classical result of E.~Calabi any hyperbolic affine hypersphere endowed with its natural Hessian metric has a non-positive Ricci tensor. The affine hyperspheres can be described as the level sets of solutions to the ``hyperbolic" toric K\"ahler-Einstein equation $e^{\Phi} = \det D^2 \Phi$ on proper convex cones. We prove a generalization of this theorem showing that for every $\Phi$ solving this equation on a proper convex domain $\Omega$ the corresponding metric measure space $(D^2 \Phi, e^{\Phi}dx)$ has a non-positive Bakry-{\'E}mery tensor. Modifying the Calabi's computations we obtain this result by applying tensorial maximum principle to the weighted Laplacian of the Bakry-{\'E}mery tensor. All of the computations are carried out in the generalized framework adapted to the optimal transportation problem for arbitrary target and source measures. For the optimal transportation of probability measures we prove a third-order uniform dimension-free a priori estimate in spirit of the second-order Caffarelli's contraction theorem.

Added: Apr 14, 2016

Working paper

We give category-theoretic reformulations of stability, NIP, NTP, and non-dividing by observing that their characterisations in terms of indiscernible sequences are naturally expressed as Quillen lifting properties of certain morphisms associated with linear orders, in a certain category extending the categories of topological spaces and of simplicial sets. This suggests an approach to a homotopy theory for model theory.

Added: Oct 29, 2020

Working paper

The classical concept of the revealed preferences was introduced by P. Samuelson and studied by H.S. Houthakker, M. Richter, S. Afriat, H. Varian and many others. It was shown by Afriat that the so called SARP (or cyclically consistence) axiom is a necessary and sufficient condition for existence of an appropriate concave utility function for a finite set of choices and prices observed (this is called the rationalization of the preferences relations by a concave utility function). Later Varian suggested some effective tests for SARP in the homogeneous case. The result on existence of a homogeneous concave utility function can be considered as a particular fact of the Monge-Kantorovich mass transportation theory, which has found numerous applications in many fields of mathematics during the last decade. In this paper we explain this viewpoint and discuss some related questions. We give an instructive and short description of the relation between these concepts, which seems somehow missing in the literature. Applying some recent results on the Monge–Kantorovich problem, we give a complete characterization of the homogeneous rationalizable data sets in the continuous case from the ”optimal mass transportation” viewpoint.

Added: Feb 23, 2013

Working paper

We study the decompositions into irreducible components of tensor products and restrictions of irreducible representations of classical Lie groups as the rank of the group goes to infinity. We prove the Law of Large Numbers for the random counting measures describing the decomposition. This leads to two operations on measures which are deformations of the notions of the free convolution and the free projection. We further prove that if one replaces counting measures with others coming from the work of Perelomov and Popov on the higher order Casimir operators for classical groups, then the operations on the measures turn into the free convolution and projection themselves.
We also explain the relation between our results and limit shape theorems for uniformly random lozenge tilings with and without axial symmetry.

Added: Dec 4, 2013

Working paper

We study the nonsymmetric Macdonald polynomials specialized at infinity from various points of view. First, we define a family of modules of the Iwahori algebra whose characters are equal to the nonsymmetric Macdonald polynomials specialized at infinity. Second, we show that these modules are isomorphic to the dual spaces of sections of certain sheaves on the semi-infinite Schubert varieties. Third, we prove that the global versions of these modules are homologically dual to the level one affine Demazure modules.

Added: Mar 20, 2017

Working paper

We explore whether a root lattice may be similar to the lattice $\mathscr O$ of integers of a number field $K$ endowed with the inner product $(x, y):={\rm Trace}_{K/\mathbb Q}(x\cdot\theta(y))$, where $\theta$ is an involution of $K$. We classify all pairs $K$, $\theta$ such that $\mathscr O$ is similar to either an even root lattice or the root lattice $\mathbb Z^{[K:\mathbb Q]}$. We also classify all pairs $K$, $\theta$ such that $\mathscr O$ is a root lattice. In addition to this, we show that $\mathscr O$ is never similar to a positive-definite even unimodular lattice of rank $\leqslant 48$, in particular, $\mathscr O$ is not similar to the Leech lattice. In appendix, we give a general cyclicity criterion for the primary components of the discriminant group of $\mathscr O$.

Added: Feb 14, 2020

Working paper

Given a connected reductive algebraic group G and a Borel subgroup B⊆G, we study B-normalized one-parameter additive group actions on affine spherical G-varieties. We establish basic properties of such actions and their weights and discuss many examples exhibiting various features. We propose a construction of such actions that generalizes the well-known construction of normalized one-parameter additive group actions on affine toric varieties. Using this construction, for every affine horospherical G-variety X we obtain a complete description of all G-normalized one-parameter additive group actions on X and show that the open G-orbit in X can be connected with every G-stable prime divisor via a suitable choice of a B-normalized one-parameter additive group action. Finally, when G is of semisimple rank 1, we obtain a complete description of all B-normalized one-parameter additive group actions on affine spherical G-varieties having an open orbit of a maximal torus T⊆B.

Added: Apr 28, 2021

Working paper

We classify the types of root systems $R$ in the rings of integers of number fields $K$ such that the Weyl group $W(R)$ lies in the group $\mathcal L(K)$ generated by ${\rm Aut} (K)$ and multipli\-ca\-tions by the elements of $K^*$. We also classify the Weyl groups of roots systems of rank $n$ which are isomorphic to a subgroup of $\mathcal L(K)$ for a number field $K$ of degree $n$ over $\mathbb Q$.

Added: Aug 8, 2018

Working paper

Given a generic family $Q$ of 2-dimensional quadrics over a smooth 3-dimensional base $Y$ we consider the relative Fano scheme $M$ of lines of it. The scheme $M$ has a structure of a generically conic bundle $M \to X$ over a double covering $X \to Y$ ramified in the degeneration locus of $Q \to Y$. The double covering $X \to Y$ is singular in a finite number of points (corresponding to the points $y \in Y$ such that the quadric $Q_y$ degenerates to a union of two planes), the fibers of $M$ over such points are unions of two planes intersecting in a point. The main result of the paper is a construction of a semiorthogonal decomposition for the derived category of coherent sheaves on $M$. This decomposition has three components, the first is the derived category of a small resolution $X^+$ of singularities of the double covering $X \to Y$, the second is a twisted resolution of singularities of $X$ (given by the sheaf of even parts of Clifford algebras on $Y$), and the third is generated by a completely orthogonal exceptional collection.

Added: Oct 4, 2013

Working paper

Added: Mar 18, 2013

Working paper

Added: Oct 18, 2019