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Of all publications in the section: 482
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Working paper
Galkin O., Galkina S. arxiv.org. math. Cornell University, 2020. No. arXiv:2012.07174.
This short communication (preprint) is devoted to mathematical study of evolution equations that are important for mathematical physics and quantum theory; we present new explicit formulas for solutions of these equations and discuss their properties. The results are given without proofs but the proofs will appear in the longer text which is now under preparation. In this paper, infinite-dimensional generalizations of the Euclidean analogue of the Schrödinger equation for anharmonic oscillator are considered in the class of measures. The Cauchy problem for these equations is solved. In particular cases, explicit formulas for fundamental solutions are obtained, which are a generalization of the Mehler formula, and the uniqueness of the solution with certain properties is proved. An analogue of the Ornstein-Uhlenbeck measure is constructed. The definition of Gaussian semigroups is given and their connection with the considered parabolic equations is described.
Working paper
Zlotnik Alexander, Zlotnik Ilya. arxiv.org. math. Cornell University, 2014. No. arXiv.org:1406.5102.
We consider the generalized time-dependent Schr\"odinger equation on the half-axis and a broad family of finite-difference schemes with the discrete transparent boundary conditions (TBCs) to solve it. We first rewrite the discrete TBCs in a simplified form explicit in space step $h$. Next, for a selected scheme of the family, we discover that the discrete convolution in time in the discrete TBC does not depend on $h$ and, moreover, it coincides with the corresponding convolution in the semi-discrete TBC rewritten similarly. This allows us to prove the bound for the difference between the kernels of the discrete convolutions in the discrete and semi-discrete TBCs (for the first time). Numerical experiments on replacing the discrete TBC convolutions by the semi-discrete one exhibit truly small absolute errors though not relative ones in general. The suitable discretization in space of the semi-discrete TBC for the higher-order Numerov scheme is also discussed.
Working paper
Ivan Shilin. arxiv.org. math. Cornell University, 2019. No. arXiv:1903.01933.
An orientation-preserving non-contractible separatrix loop of a hyperbolic saddle of a vector field on a two-dimensional surface may be accumulated by a separatrix of the same saddle. We study the unfolding of such loops in generic one-parameter families of vector fields as a semi-local bifurcation. As a byproduct, we construct a countable family of pairwise non-equivalent germs of bifurcation diagrams that appear in locally generic one-parameter families.
Working paper
Tyurin N. A. arxiv.org. math. Cornell University, 2015
We present a new approach to special lagrangian geometry which works for Bohr - Sommerfeld lagrangian submanifolds of symplectic manifolds with integer symplectic forms. This leads to construction of finite dimensional moduli spaces of SBS lagrangian cycles over algebraic varieties.
Working paper
Tyurin N. A. arxiv.org. math. Cornell University, 2010. No. 1005.2006.
Working paper
Pavel S. Prudnikov. arxiv.org. math. Cornell University, 2020
Paul Chernoff in 1968 proposed his approach to approximations of one-parameter operator semigroups while trying to give a rigorous mathematical meaning to Feynman's path integral formulation of quantum mechanics. In early 2000's Oleg Smolyanov noticed that Chernoff's theorem may be used to obtain approximations to solutions of initial-value problems for linear partial differential equations (LPDEs) of evolution type with variable coefficients, including parabolic equations, Schr\"odinger equation, and some other. Chernoff expressions are explicit formulas containing variable coefficients of LPDE and the initial condition, hence they can be used as a numerical method for solving LPDEs. However, the speed of convergence of such approximations at the present time is understudied which makes it risky to employ this class of numerical methods. In the present paper we take two equations with known solutions (heat equation and transport equation) and study both analytically and numerically the speed of decay of the norm of the difference between Chernoff approximations and exact solutions. We also provide graphical illustrations of convergence and its rate. These model examples, being relatively simple, allow to demonstrate general properties of Chernoff approximations. The observations obtained build a base for the future employment of the approach based on Chernoff's theorem to the problem of construction of new numerical methods for solving initial-value problem for parabolic LPDEs with variable coefficients.
Working paper
Fedor Bogomolov, Böhning C. arxiv.org. math. Cornell University, 2012
In this article we determine the stable cohomology groups H^i_s (A_n, Z/p) of the alternating groups A_n for all integers n and i, and all primes p.
Working paper
Soldatenkov A. O., Verbitsky M. arxiv.org. math. Cornell University, 2012. No. 1202.0222v1.

A hypercomplex manifold M is a manifold with a triple I,J,K of complex structure operators satisfying quaternionic relations. For each quaternion L=aI +bJ+cK, L^2=-1, L is also a complex structure operator on M, called an induced complex structure. We are studying compact complex subvarieties of (M,L), when L is a generic induced complex structure. Under additional assumptions (Obata holonomy contained in SL(n,H), existence of an HKT metric), we prove that (M,L) contains no divisors, and all complex subvarieties of codimension 2 are trianalytic (that is, also hypercomplex).

Working paper
Abramov Y. V. arxiv.org. math. Cornell University, 2011. No. arXiv:1111.4974v1.

I give the explicit formula for the (set-theoretical) system of Resultants of m+1 homogeneous polynomials in n+1 variables

Working paper
Netay I. V. arxiv.org. math. Cornell University, 2016
In this work we explicitly calculate syzygies of quadratic Veronese embedding~$\mathbb{P}(V)\subset\mathbb{P}(\Sym^2V)$ as representations of the group~$\GL(V)$. Also resolutions of the sheaves $\mathscr{O}_{\mathbb{P}(V)}(i)$ are constructed in the category~$D(\mathbb{P}(\Sym^2V))$.
Working paper
Netay I. V. arxiv.org. math. Cornell University, 2011
We describe the syzygy spaces for the Segre embedding $\mathbb{P}(U)\times\mathbb{P}(V)\subset\mathbb{P}(U\otimes V)$ in terms of representations of $\GL(U)\times \GL(V)$ and construct the minimal resolutions of the sheaves $\mathscr{O}_{\mathbb{P}(U)\times\mathbb{P}(V)}(a,b)$ in $D(\mathbb{P}(U\otimes V))$ for $a\geqslant-\dim(U)$ and $b\geqslant-\dim(V)$. Also we prove some property of multiplication on syzygy spaces of the Segre embedding.
Working paper
Vladimir Lebedev. arxiv.org. math. Cornell University, 2018. No. arXiv:1806.06386v2.
We give a complete characterization of tame semicascades and cascades generated by affine self-mappings os d-torus.
Working paper
Verbitsky M. arxiv.org. math. Cornell University, 2014
A Teichm\"uller space Teich is a quotient of the space of all complex structures on a given manifold M by the connected components of the group of diffeomorphisms. The mapping class group Γ of M is the group of connected components of the diffeomorphism group. The moduli problems can be understood as statements about the Γ-action on Teich. I will describe the mapping class group and the Teichmuller space for a hyperkahler manifold. It turns out that this action is ergodic. We use the ergodicity to show that a hyperkahler manifold is never Kobayashi hyperbolic. This is my ICM submission, with review of some of my work on Teichmuller spaces and moduli; proofs are sketched, new observations and some open problems added.
Working paper
Marcati C., Rakhuba M., Christoph S. arxiv.org. math. Cornell University, 2020
We analyze rates of approximation by quantized, tensor-structured representations of functions with isolated point singularities in ℝ3. We consider functions in countably normed Sobolev spaces with radial weights and analytic- or Gevrey-type control of weighted semi-norms. Several classes of boundary value and eigenvalue problems from science and engineering are discussed whose solutions belong to the countably normed spaces.  It is shown that quantized, tensor-structured approximations of functions in these classes exhibit tensor ranks bounded polylogarithmically with respect to the accuracy ϵ∈(0,1) in the Sobolev space H1. We prove exponential convergence rates of three specific types of quantized tensor decompositions: quantized tensor train (QTT), transposed QTT and Tucker-QTT. In addition, the bounds for the patchwise decompositions are uniform with respect to the position of the point singularity. An auxiliary result of independent interest is the proof of exponential convergence of hp-finite element approximations for Gevrey-regular functions with point singularities in the unit cube Q=(0,1)3. Numerical examples of function approximations and of Schrödinger-type eigenvalue problems illustrate the theoretical results.
Working paper
Burmistrova E., Lobanov S. G. arxiv.org. math. Cornell University, 2018
This note proves that the representation of the Allen elasticity of substitution obtained by Uzawa for linear homogeneous functions holds true for nonhomogeneous functions. It is shown that the criticism of the Allen-Uzawa elasticity of substitution in the works of Blackorby, Primont, Russell is based on an incorrect example.
Working paper
F.A. Bogomolov, Vik.S. Kulikov. arxiv.org. math. Cornell University, 2014
In \cite{Ku0}, the ambiguity index $a_{(G,O)}$ was introduced for each equipped finite group $(G,O)$. It is equal to the number of connected components of a Hurwitz space parametrizing coverings of a projective line with Galois group $G$ assuming that all local monodromies belong to conjugacy classes $O$ in $G$ and the number of branch points is greater than some constant. We prove in this article that the ambiguity index can be identified with the size of a generalization of so called Bogomolov multiplier (\cite{Kun1}, see also \cite{BO87}) and hence can be easily computed for many pairs $(G,O)$.
Working paper
Pirkovskii A. Y. arxiv.org. math. Cornell University, 2011. No. 1101.0166.
Given a Hopf algebra H and an H-module algebra A, we explicitly describe the Arens-Michael envelope of the smash product A#H in terms of the Arens-Michael envelope of H and a certain completion of A. We also give an example (Manin's quantum plane) showing that the result fails for non-Hopf bialgebras.
Working paper
Olshanski G., Borodin A. arxiv.org. math. Cornell University, 2016
We introduce a family of discrete determinantal point processes related to orthogonal polynomials on the real line, with correlation kernels defined via spectral projections for the associated Jacobi matrices. For classical weights, we show how such ensembles arise as limits of various hypergeometric orthogonal polynomials ensembles. We then prove that the q-Laplace transform of the height function of the ASEP with step initial condition is equal to the expectation of a simple multiplicative functional on a discrete Laguerre ensemble --- a member of the new family. This allows us to obtain the large time asymptotics of the ASEP in three limit regimes: (a) for finitely many rightmost particles; (b) GUE Tracy-Widom asymptotics of the height function; (c) KPZ asymptotics of the height function for the ASEP with weak asymmetry. We also give similar results for two instances of the stochastic six vertex model in a quadrant. The proofs are based on limit transitions for the corresponding determinantal point processes.
We study a special type of $E_\infty$-operads that govern strictly unital $E_\infty$-coalgebras (and algebras) over the ring of integers. Morphisms of coalgebras over such an operad are defined by using universal $E_\infty$-bimodules. Thus we obtain a category of $E_\infty$-coalgebras. It turns out that if the homology of an $E_\infty$-coalgebra have no torsion, then there is a natural way to define an $E_\infty$-coalgebra structure on the homology so that the resulting coalgebra be isomorphic to the initial $E_\infty$-coalgebra in our category. We also discuss some invariants of the $E_\infty$-coalgebra structure on homology and relate them to the invariant formerly used by the author to distinguish the fundamental groups of the complements of combinatorially equivalent complex hyperplane arrangements.